Algebra Quiz: Function Notation And Evaluation

20 questions · exam conditions

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Function Notation And EvaluationQuestion 1 of 20

A plant grows at a steady rate. Its height (in centimeters) after $t$ weeks is $H(t) = 3t + 10$ . What is $H(4)$ ?

22

17

42

13

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Algebra Quiz

Algebra Quiz: Function Notation And Evaluation

Practice Function Notation And Evaluation in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Function Notation And Evaluation, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A plant grows at a steady rate. Its height (in centimeters) after $t$ weeks is $H(t) = 3t + 10$ . What is $H(4)$ ?

$22$ (correct answer)
$17$
$42$
$13$

H(4) = 3(4) + 10 = 12 + 10 = 22

centimeters, which means the height after 4 weeks is 22 centimeters. The function helps us quickly answer 'what if' questions by just plugging in different input values! Choice A is correct because it properly substitutes 4 for t in the function and calculates accurately:

3(4) + 10 = 22

. Nice work if you got this! Choice C is a common slip-up: it makes an arithmetic error, calculating

3*4 + 10

12 + 5

or something similar. Double-checking your arithmetic is always a good idea! In word problems, always state what your answer means: don't just write 'H(4) = 22'—say 'H(4) = 22 centimeters, which is the height after 4 weeks.' This shows you understand what the math represents!

Question 2

If $m(x)=2x^2-3x+1$ , evaluate $m(-1)$ .

$0$
$2$
$4$
$6$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like m(x) = 2x² - 3x + 1 at a specific value, we replace every x with that value and calculate: m(-1) means substitute -1 for x, giving 2(-1)² - 3(-1) + 1. When evaluating with a negative number like m(-1), we need to be extra careful with signs! Substituting -1 for x in m(x) = 2x² - 3x + 1, we get m(-1) = 2(-1)² - 3(-1) + 1, which equals 2(1) + 3 + 1 = 2 + 3 + 1 = 6. Notice how the parentheses around -1 help keep track of the negative! Choice D is correct because it properly substitutes -1 for x in the function and calculates accurately: 2(-1)² - 3(-1) + 1 = 2(1) + 3 + 1 = 6. Nice work if you got this! Choice A (0) might result from sign errors, possibly calculating 2(-1) - 3(-1) + 1 without squaring the first term. Remember to follow the function formula exactly as written! When substituting negative numbers, always use parentheses to protect yourself from sign errors: write m(-1) = 2(-1)² - 3(-1) + 1. The parentheses keep everything clear! Remember that (-1)² = 1 and -3(-1) = +3, so negative inputs can lead to positive terms!

Question 3

The cost in dollars to rent a bike for $x$ hours is $C(x)=8x+12$ . What is $C(3)$ ?

$36$ (correct answer)
$20$
$8$
$24$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. In real-world contexts, a function like C(x) = 8x + 12 gives us a formula to calculate one quantity (like cost) from another (like number of hours), and evaluating C(3) tells us the specific cost when 3 hours are involved. Starting with C(x) = 8x + 12 and finding C(3), we substitute 3 for x everywhere: C(3) = 8(3) + 12. Now we calculate step by step: 8(3) + 12 = 24 + 12 = 36. In this problem, C(3) = 36 dollars, which means the cost to rent a bike for 3 hours is $36. The function helps us quickly answer 'what if' questions by just plugging in different input values! Choice B is correct because it properly substitutes 3 for x in the function and calculates accurately: 8(3) + 12 = 36. Nice work if you got this! Choice C has the right idea but makes an arithmetic error, calculating 8(3) = 24 but then forgetting to add the 12. Double-checking your arithmetic is always a good idea! In word problems, always state what your answer means: don't just write 'C(3) = 36'—say 'C(3) = 36 dollars, which is the total cost to rent a bike for 3 hours.' This shows you understand what the math represents!

Question 4

For $f(x)=4-2x$ , what is the value of $f(0)$ ?

$-2$
$0$
$2$
$4$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like f(x) = 4 - 2x at a specific value, we replace every x with that value and calculate: f(0) means substitute 0 for x, giving 4 - 2(0). Starting with f(x) = 4 - 2x and finding f(0), we substitute 0 for x everywhere: f(0) = 4 - 2(0). Now we calculate step by step: 2(0) = 0, then 4 - 0 = 4. Choice D is correct because it properly substitutes 0 for x in the function and calculates accurately: 4 - 2(0) = 4 - 0 = 4. Nice work if you got this! Choice B (0) treats f(0) as multiplication (f times 0), but function notation uses parentheses to show the input, not to multiply. Think of f as a 'machine' that processes inputs, not as a number to multiply with. Here's the foolproof way to evaluate functions: (1) write out the function formula, (2) wherever you see the variable, write the input value in parentheses, (3) calculate step by step using order of operations. When the input is 0, any term with x becomes 0, making the calculation especially simple!

Question 5

A taxi charges a base fee plus a per-mile fee. The total cost (in dollars) for $x$ miles is $C(x) = 4x + 3$ . What is $C(5)$ ?

$12$
$35$
$20$
$23$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. In real-world contexts, a function like C(x) = 4x + 3 gives us a formula to calculate one quantity (like cost) from another (like number of miles), and evaluating C(5) tells us the specific cost when 5 miles are involved. In this problem, C(5) = 4(5) + 3 = 20 + 3 = 23 dollars, which means the total cost for 5 miles is 23 dollars. The function helps us quickly answer 'what if' questions by just plugging in different input values! Choice A is correct because it properly substitutes 5 for x in the function and calculates accurately: 4(5) + 3 = 23. Nice work if you got this! Choice B is a common slip-up: it makes an arithmetic error, calculating 4*5 + 3 as 20 without adding the 3. Double-checking your arithmetic is always a good idea! In word problems, always state what your answer means: don't just write 'C(5) = 23'—say 'C(5) = 23 dollars, which is the total cost for 5 miles.' This shows you understand what the math represents!

Question 6

A movie theater charges a base fee plus a per-ticket cost. The total cost in dollars for $x$ tickets is $C(x)=9x+6$ . What is $C(3)$ ?

$21$
$27$
$33$ (correct answer)
$45$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. In real-world contexts, a function like C(x) = 9x + 6 gives us a formula to calculate one quantity (like cost) from another (like number of tickets), and evaluating C(3) tells us the specific cost when 3 tickets are involved. Starting with C(x) = 9x + 6 and finding C(3), we substitute 3 for x everywhere: C(3) = 9(3) + 6. Now we calculate step by step: 9(3) = 27, then 27 + 6 = 33. Choice C is correct because it properly substitutes 3 for x in the function and calculates accurately: 9(3) + 6 = 27 + 6 = 33. Nice work if you got this! Choice B (27) only calculates 9(3) = 27 without adding the base fee of 6. Remember to include all parts of the function when evaluating! In this problem, C(3) = 33 dollars, which means the total cost for 3 movie tickets is $33. The function helps us quickly answer 'what if' questions by just plugging in different input values! In word problems, always state what your answer means: don't just write 'C(3) = 33'—say 'C(3) = 33 dollars, which is the total cost for 3 tickets.' This shows you understand what the math represents!

Question 7

A parking garage charges based on the function $P(t) = 5 + 2⌊t⌋$ where $t$ is time in hours and $⌊t⌋$ is the greatest integer less than or equal to $t$ .

Sarah parks for 2.7 hours. If the equation $P(2.7) = c$ represents her total cost, what does the expression $P(2.7) - P(1.3)$ represent?

The difference in parking time between Sarah's visit and a 1.3-hour visit
The total cost for parking 1.4 hours at the garage's standard rate
The average hourly rate Sarah pays for the extra 1.4 hours of parking
The additional cost Sarah pays compared to someone who parks for 1.3 hours (correct answer)

Explanation: When you encounter function notation with operations between two function values, focus on what each piece represents and how their combination creates meaning in the real-world context. Let's calculate each value using

P(t) = 5 + 2⌊t⌋

. For Sarah's 2.7-hour visit:

P(2.7) = 5 + 2⌊2.7⌋ = 5 + 2(2) = 9

dollars. For a 1.3-hour visit:

P(1.3) = 5 + 2⌊1.3⌋ = 5 + 2(1) = 7

dollars. Therefore,

P(2.7) - P(1.3) = 9 - 7 = 2

dollars. This difference represents how much more Sarah pays compared to someone parking for 1.3 hours, making D correct. Choice A confuses the mathematical operation with what's being measured. The expression

P(2.7) - P(1.3)

involves costs, not time differences. While the time difference is indeed

2.7 - 1.3 = 1.4

hours, that's not what this expression calculates. Choice B misinterprets the result. The $2 difference isn't the cost for parking 1.4 hours total—it's the additional cost between two different parking durations. Choice C incorrectly suggests we're finding an average rate. An average would require dividing by time, but this expression only subtracts costs. Study tip: When you see function differences like $

f(a) - f(b)

$, always ask "What does this tell me about the relationship between the two scenarios?" The difference between function outputs typically represents how much one situation differs from another in terms of the function's measured quantity.

Question 8

A delivery service charges according to the function $C(w) = 12 + 3w$ where $w$ is the weight in pounds and $C(w)$ is the cost in dollars.

If a customer pays $27 for shipping, and the statement "$ $C(w) = 27$ $" represents this situation, what does the expression "$ $C(w + 2)$ $" represent in this context?

The cost to ship a package that weighs 2 pounds more than the original package (correct answer)
The cost to ship the original package plus an additional $2 service fee
The total cost for shipping two packages of the same weight as the original
The cost per pound for a package that weighs 2 pounds more than the original

Explanation:

C(w + 2)

means we substitute

(w + 2)

for

w

in the original function, giving

C(w + 2) = 12 + 3(w + 2)

. This represents the cost for a package that weighs 2 pounds more than the weight

w

. Choice B incorrectly interprets the

+2

as an additional fee rather than additional weight. Choice C misunderstands this as shipping multiple packages. Choice D incorrectly suggests this gives a per-pound rate rather than a total cost.

Question 9

For a function $p(x)$ , it is known that $p(5) = 8$ , $p(8) = 2$ , and $p(2) = 5$ . What is the value of $p(p(p(5)))$ ?

$2$
$5$ (correct answer)
$8$
$15$

Explanation: We evaluate from the inside out. First,

p(5) = 8

. Then

p(p(5)) = p(8) = 2

. Finally,

p(p(p(5))) = p(2) = 5

. Choice A (

2

) represents stopping at

p(p(5))

, Choice C (

8

) represents stopping at

p(5)

, and Choice D (

15

) represents incorrectly adding the intermediate values.

Question 10

Let $f(x) = 2x - 1$ and $g(x) = x^2 + 3$ . If $f(a) = 9$ and $g(b) = 12$ , what is the value of $f(b) + g(a)$ ?

$27$
$42$
$36$
$33$ (correct answer)

Explanation: This problem tests function evaluation and substitution—core skills where you plug values into functions and work with the results strategically. Start by finding the values of

a

and

b

. Since

f(a) = 9

, substitute into

f(x) = 2x - 1

2a - 1 = 9

, so

2a = 10

and

a = 5

. Since

g(b) = 12

, substitute into

g(x) = x^2 + 3

b^2 + 3 = 12

, so

b^2 = 9

and

b = 3

(taking the positive solution). Now calculate

f(b) + g(a)

. With

b = 3

f(3) = 2(3) - 1 = 5

. With

a = 5

g(5) = 5^2 + 3 = 28

. Therefore,

f(b) + g(a) = 5 + 28 = 33

. Choice A (

27

) likely comes from incorrectly calculating

g(5) = 25 - 1 + 3 = 27

, mixing up the function formulas. Choice B (

42

) might result from computing

f(5) + g(3) = 9 + 12 = 21

and doubling it, or from other calculation errors. Choice C (

36

) could come from finding

a = 5

and

b = 3

correctly but then computing

ab + f(b) = 15 + 5 + 16 = 36

, confusing what expression to evaluate. The key strategy: solve for the unknown inputs first, then carefully substitute into the requested expression. Always double-check which function you're applying to which variable—it's easy to mix up

f(a)

versus

f(b)

when working quickly.

Question 11

A function $m(x)$ satisfies $m(x + 3) = 2m(x) + 1$ for all values in its domain. If $m(1) = 4$ , what is $m(7)$ ?

$9$
$17$
$19$ (correct answer)
$35$

Explanation: We use the recursive relationship

m(x + 3) = 2m(x) + 1

. Starting with

m(1) = 4

m(4) = m(1 + 3) = 2m(1) + 1 = 2(4) + 1 = 9

. Then

m(7) = m(4 + 3) = 2m(4) + 1 = 2(9) + 1 = 19

. Choice A represents

m(4)

only, Choice B represents forgetting to add 1 in the final step, and Choice D represents incorrectly multiplying by 3 instead of applying the relationship twice.

Question 12

Given $h(x) = x^2 + 1$ , what is $h(x + 2)$ ?

$x^2 + 3$
$(x+2)^2 + 1$ (correct answer)
$x^2 + 4x + 1$
$x^2 + 2x + 3$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation h(x) tells us the rule for calculating outputs from inputs: when you see h(x + 2), it means 'substitute (x + 2) for every x in the function formula,' like filling in a blank everywhere you see x. For h(x + 2), we substitute the entire expression (x + 2) for every x: h(x + 2) = (x + 2)² + 1. This gives us (x + 2)² + 1, which is the expression as is. Choice B is correct because it properly substitutes (x + 2) for x in the function: (x + 2)² + 1. Nice work if you got this! Choice C is a common slip-up: it expands the expression to x² + 4x + 4 + 1 = x² + 4x + 5 but the question asks for h(x + 2) without specifying to expand, though B keeps it in the substituted form. Remember, sometimes the answer is the substituted expression itself! To check your work, ask yourself: 'Did I replace every single x with my input value?' and 'Did I follow PEMDAS for my calculation?' These two checks catch most mistakes!

Question 13

Given $g(x) = 3x^2 + 2x - 1$ , what is $g(-1)$ ?

$0$ (correct answer)
$-6$
$4$
$2$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like

g(x) = 3x^2 + 2x - 1

at a specific value, we replace every x with that value and calculate:

g(-1)

means substitute -1 for x, giving

3(-1)^2 + 2(-1) - 1 = 3(1) - 2 - 1 = 0

. When evaluating with a negative number like

g(-1)

, we need to be extra careful with signs! Substituting -1 for x in

g(x) = 3x^2 + 2x - 1

, we get

3(-1)^2 + 2(-1) - 1

, which equals

3(1) - 2 - 1 = 0

. Notice how the parentheses around -1 help keep track of the negative! Choice A is correct because it properly substitutes -1 for x in the function and calculates accurately:

3 - 2 - 1 = 0

. Nice work if you got this! Choice C is a common slip-up: it makes a sign error, perhaps calculating

3(-1)^2

3(-1) = -3

or mishandling the signs. When you substitute a negative number, use parentheses to keep track:

(-1)^2

is positive, but forgetting can lead to errors. When substituting negative numbers, always use parentheses to protect yourself from sign errors: write

f(-3) = 2(-3)^2 + 1

, not

f(-3) = 2-3^2 + 1

. The parentheses keep everything clear!

Question 14

For $k(x)=x^2-1$ , find both $k(3)$ and $k(-3)$ .

$k(3)=8$ and $k(-3)=-8$
$k(3)=-8$ and $k(-3)=-8$
$k(3)=8$ and $k(-3)=8$ (correct answer)
$k(3)=-8$ and $k(-3)=8$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation f(x) tells us the rule for calculating outputs from inputs: when you see f(3), it means 'substitute 3 for every x in the function formula,' like filling in a blank everywhere you see x. When evaluating with a negative number like k(-3), we need to be extra careful with signs! Substituting -3 for x in k(x) = x² - 1, we get (-3)² - 1 = 9 - 1 = 8, and for k(3) = (3)² - 1 = 9 - 1 = 8. Notice how the parentheses around -3 help keep track of the negative! Choice A is correct because it properly substitutes both 3 and -3 for x and calculates accurately: both give 8. Nice work if you got this! Choice B is a common slip-up: it makes a sign error when working with the negative input -3, perhaps thinking (-3)² is -9 instead of 9. When you substitute a negative number, use parentheses to keep track: (-3)² is positive, but forgetting parentheses can lead to the wrong sign. When substituting negative numbers, always use parentheses to protect yourself from sign errors: write k(-3) = (-3)² - 1, not -3² - 1. The parentheses keep everything clear!

Question 15

Given the piecewise function $f(x)=\begin{cases}2x+1, & x\ge 0\\ -x+4, & x<0\end{cases}$ what is $f(-3)$ ?

$-5$
$-2$
$7$ (correct answer)
$3$

Explanation: This question tests your understanding of function notation and how to evaluate piecewise functions by choosing the correct formula based on the input value. Function notation f(x) with a piecewise definition means we have different rules for different input values: when you see f(-3), first check which condition -3 satisfies, then use that formula. Since -3 < 0, we use the second formula: f(x) = -x + 4. Substituting -3 for x, we get f(-3) = -(-3) + 4 = 3 + 4 = 7. Notice how -(-3) becomes positive 3! Choice C is correct because it properly identifies that -3 < 0, uses the correct formula, and calculates accurately: -(-3) + 4 = 7. Nice work if you got this! Choice D makes a sign error when working with the negative input -3, calculating -(-3) as -3 instead of +3. When you have a negative of a negative, it becomes positive! Think of a piecewise function as a machine with different settings: you check which setting to use based on your input (is -3 ≥ 0 or < 0?), then apply that specific rule. Always check your condition first before substituting!

Question 16

If $s(x)=x^2+2$ , what is $s(-4)$ ?

$-14$
$18$ (correct answer)
$-18$
$14$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation s(x) tells us the rule for calculating outputs from inputs: when you see s(-4), it means 'substitute -4 for every x in the function formula,' like filling in a blank everywhere you see x. When evaluating with a negative number like s(-4), we need to be extra careful with signs! Substituting -4 for x in s(x) = x² + 2, we get s(-4) = (-4)² + 2, which equals 16 + 2 = 18. Notice how the parentheses around -4 help us remember that (-4)² = 16 (positive!). Choice B is correct because it properly substitutes -4 for x in the function and calculates accurately: (-4)² = 16, then 16 + 2 = 18. Nice work if you got this! Choice C gives -18, which would result from incorrectly thinking (-4)² = -16 instead of +16. Remember, a negative number squared is always positive! When substituting negative numbers, always use parentheses to protect yourself from sign errors: write s(-4) = (-4)² + 2, not s(-4) = -4² + 2. The parentheses make all the difference!

Question 17

Given $p(x)=4(x-2)$ , what is the value of $p(3)$ ?

$4$ (correct answer)
$12$
$-4$
$1$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation p(x) tells us the rule for calculating outputs from inputs: when you see p(3), it means 'substitute 3 for every x in the function formula,' like filling in a blank everywhere you see x. Starting with p(x) = 4(x - 2) and finding p(3), we substitute 3 for x everywhere: p(3) = 4(3 - 2). Now we calculate step by step: first the parentheses 3 - 2 = 1, then multiply by 4 to get 4(1) = 4. Choice A is correct because it properly substitutes 3 for x in the function and calculates accurately: 4(3 - 2) = 4(1) = 4. Nice work if you got this! Choice B gives 12, which is a common slip-up: it multiplies 4 times 3 first to get 12, then subtracts 2 to get 10, or distributes incorrectly. Remember to handle what's inside parentheses first! Here's the foolproof way to evaluate functions: (1) write out the function formula, (2) wherever you see the variable, write the input value in parentheses, (3) calculate step by step using order of operations. Always do operations inside parentheses before multiplying!

Question 18

For $h(x) = (x + 5)^2$ , what is $h(-5)$ ?

$-25$
$10$
$25$
$0$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation h(x) tells us the rule for calculating outputs from inputs: when you see h(-5), it means 'substitute -5 for every x in the function formula,' like filling in a blank everywhere you see x. When evaluating with a negative number like h(-5), we need to be extra careful with signs! Substituting -5 for x in h(x) = (x + 5)², we get (-5 + 5)², which equals (0)² = 0. Notice how the parentheses around -5 help keep track of the negative! Choice A is correct because it properly substitutes -5 for x in the function and calculates accurately: (0)² = 0. Nice work if you got this! Choice C is a common slip-up: it calculates without the parentheses, like -5 + 5² = -5 + 25 = 20, but that's not the right order. When substituting negative numbers, always use parentheses to protect yourself from sign errors: write f(-3) = 2(-3)² + 1, not f(-3) = 2-3² + 1. The parentheses keep everything clear!

Question 19

Based on the graph shown, which statement about function notation is correct?

$k(-2) > k(1)$ and $k(0) = k(3)$
$k(-2) < k(1)$ and $k(0) > k(3)$
$k(-2) = k(1)$ and $k(0) < k(3)$
$k(-2) < k(1)$ and $k(0) = k(3)$ (correct answer)

Explanation: From the graph:

k(-2) = 1

k(1) = 3

k(0) = 2

, and

k(3) = 2

. Therefore

k(-2) < k(1)

(since

1 < 3

) and

k(0) = k(3)

(since both equal 2). Choice A incorrectly states

k(-2) > k(1)

, Choice B incorrectly states

k(0) > k(3)

, and Choice C incorrectly states

k(-2) = k(1)

and

k(0) < k(3)

Question 20

Given the function $f(x)=2x-7$ , what is $f(4)$ ?

$-15$
$1$ (correct answer)
$8$
$15$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like f(x) = 2x - 7 at a specific value, we replace every x with that value and calculate: f(4) means substitute 4 for x, giving 2(4) - 7 = 8 - 7 = 1. Starting with f(x) = 2x - 7 and finding f(4), we substitute 4 for x everywhere: f(4) = 2(4) - 7. Now we calculate step by step: 2(4) = 8, then 8 - 7 = 1. Choice B is correct because it properly substitutes 4 for x in the function and calculates accurately: 2(4) - 7 = 8 - 7 = 1. Nice work if you got this! Choice A (-15) makes a sign error, possibly calculating -2(4) - 7 instead of 2(4) - 7. Remember to carefully follow the signs given in the function! Here's the foolproof way to evaluate functions: (1) write out the function formula, (2) wherever you see the variable, write the input value in parentheses, (3) calculate step by step using order of operations. For example: f(x) = 3x - 2, find f(4) → f(4) = 3(4) - 2 = 12 - 2 = 10. Easy!