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SAT Math : SAT Mathematics

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Example Questions

Example Question #134 : Algebraic Functions

Function 4

Define as the function graphed above. Define function $g(x) = x^{2}- 11$ .

Evaluate $\left (f \circ g \right ) (4)$ .

Possible Answers:

$\left (f \circ g \right ) (4) = -15$

4 is outside the domain of $f \circ g$

$\left (f \circ g \right ) (4) = - \frac{1}{2}$

$\left (f \circ g \right ) (4) = -10$

$\left (f \circ g \right ) (4) = -7$

Correct answer:

$\left (f \circ g \right ) (4) = - \frac{1}{2}$

Explanation:

$\left (f \circ g \right ) (4) = f \left [ g (4)\right ]$

$g(x) = x^{2}- 11$ , so

$g(4) = 4^{2}- 11 = 16-11 = 5$

$\left (f \circ g \right ) (4) = f (5)$ .

From the diagram below, we see that $f(5) = -\frac{1}{2}$ .

Function 4a

The correct response is that $\left (f \circ g \right ) (4) = - \frac{1}{2}$ .

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Example Question #1061 : Algebra

Two functions

Define and to be the functions graphed above. For which of the following values of is the statement

(f-g)(x) = 0

a true statement?

Possible Answers:

x= 0

x= -1

x= -2

x=1

The statement is not correct for any value of .

Correct answer:

x= 0

Explanation:

f(x) - g(x) = 0

f(x)= g(x)

This can be solved by graphing and on the same set of axes and noting their points of intersection:

Two functions together

The graphs of the two functions intersect at the point (0, 1) . Therefore,

f(0) = g(0) , and

(f-g)(0) = 0 .

The correct response is x= 0 .

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Example Question #1061 : Algebra

Function 4

Define to be the function graphed above.

Which of the following is an -intercept of the graph of the function , if is defined as

g(x ) = f(x+ 5) ?

Possible Answers:

The graph of has no -intercept.

$\left ( -4\frac{1}{2}, 0 \right )$

(-9, 0)

(-7, 0)

(-3, 0)

Correct answer:

(-7, 0)

Explanation:

An -intercept of the graph of has as its -coordinate a value such that

g(x) = 0 ,

or, equivalently,

f(x+ 5) = 0

From the diagram, we can see that

$f(-8) = f(-2) = f \left ( 5 \frac{1}{2}\right ) = 0$

Therefore, to find the -intercepts of , set x+5 equal to these numbers; equivalently, subtract 5 from each number. We get that

$g(-13) = g(-7) = g \left ( \frac{1}{2} \right )= 0$

Therefore, the -intercepts of the graph of are the points

(-13, 0) , (-7, 0) , $\left ( \frac{1}{2}, 0 \right )$ .

The correct choice is (-7, 0) .

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Example Question #136 : Algebraic Functions

Function 4

Define to be the function graphed above.

Give the -intercept of the graph of the function , which is defined as

g(x) = f(x-4) .

Possible Answers:

(0,4)

(0,0)

(0,-8)

The graph of has no -intercept.

(0, -2)

Correct answer:

(0,4)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = f(x-4)

g(0) = f(0-4)= f( -4)

As can be seen in the diagram below, f( -4) = 4 .

Function 4a

Therefore, g(0) = 4 , and the correct response is (0,4) .

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Example Question #137 : Algebraic Functions

Define a function as follows:

$g (x) = x ^{2} - 8x + 17$ , where the domain of is the set [0, 9] .

Give the range of .

Possible Answers:

[17, 26]

$[17, \infty )$

[1, 26]

$[1 , \infty )$

[1, 17]

Correct answer:

[17, 26]

Explanation:

This problem can be solved by examining the behavior of the graph of $g (x) = x ^{2} - 8x + 17$ , which is a parabola.

Since the quadratic coefficient is 1, a positive number, its vertex is a minumum. The -coordinate of the vertex can be found by setting a = 1, b= -8 , and calculating:

$x = - \frac{b}{2a} = -\frac{-8}{2 \cdot 1} = 4$

The -coordinate is

$g (4) = 4 ^{2} - 8 (4) + 17 = 16 - 32 + 17 = 1$

The minimum value of is therefore 1, and this occurs at $x = 4 \in [0, 9]$ . This makes 1 the lower bound of the range.

Since the graph of is a parabola, it decreases everywhere for x < 4 and increases everywhere for x > 4 . Therefore, we can evaluate g(0) and g(9) , and choose the higher value as the maximum value on the given domain.

$g (0) = 0 ^{2} - 8 (0)+ 17 = 0 - 0 + 17 = 17$

$g (0) = 9 ^{2} - 8 (9)+ 17 = 81 - 72 + 17 = 26$

We choose 26 as the upper bound of the range.

Therefore, the range of , given the domain restriction, is the set [17, 26] .

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Example Question #1061 : Algebra

$\frac{x-4}{x}=\frac{6}{18}$

Possible Answers:

Correct answer:

Explanation:

The first step is to cross multiple which leaves you with 18x-72=6x . The next step is to get all x's on one side of the equation and all constants on the other. You can add to both sides then subtract from both sides. This gives us 12x=72 . The last step is to get by itself by dividing each side by giving an answer of x=6

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Example Question #142 : Algebraic Functions

$f(x) = \left | 15-x^{2}\right |$ If f(x) = 10 , what is the positive value of that satisfies the equation?

Possible Answers:

Correct answer:

Explanation:

Since the final answer is with the absolute value, then $15-x^{2}$ can be or . To get , you have to plug in $\sqrt{5}$ for $(15-\sqrt[2]{5}=10)$ . To get , you plug in $(15-5^{2}=10)$ .

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Example Question #143 : Algebraic Functions

Find f(x) : $3x-\frac{6}{f(x)}=8$

Possible Answers:

$f(x)=\frac{6}{8-3x}$

$f(x)=\frac{6}{3x-8}$

$f(x)= \frac{1}{3}x+\frac{4}{3}$

$f(x) = -\frac{1}{2}x+4$

$f(x) = \frac{1}{2}x+4$

Correct answer:

$f(x)=\frac{6}{3x-8}$

Explanation:

To simplify this equation, first subtract on both sides of the equation.

$-\frac{6}{f(x)}=8-3x$

Multiply f(x) on both sides of the equation to eliminate the denominator on the left side of the equation.

$-\frac{6}{f(x)}=8-3x$

-6=f(x)(8-3x)

Divide by (8-3x) on both sides of the equation to isolate f(x) .

$\frac{-6}{(8-3x)}=f(x)$

The negative sign can be extracted out in front of the fraction.

$f(x)=-\frac{6}{8-3x}$

The negative sign can then be distributed with the denominator.

$f(x)=\frac{6}{-(8-3x)} = \frac{6}{3x-8}$

The correct answer is: $f(x)=\frac{6}{3x-8}$

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Example Question #2702 : Act Math

For which value of are the following two functions equal?

$F(x)=2x+3^x+\frac{9x}{3}$

$G(x)=\frac{2^x+4^x}{2}-4x-5x+1$

Possible Answers:

Correct answer:

Explanation:

It is important to follow the order of operations for this equation and find a solution that satisfies both F(x) and G(x).

Recall the order of operations is PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

The correct answer is 4 because

F(x) = 2x + 3^x+ (9x/3) = 2(4) + 3⁴ + ((9 * 4)/3) = 101, and

G(x) = (((2⁴ + 4⁴)/2) - 4 * 4) – 5(4) + 1 = 101.

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Example Question #141 : Algebraic Functions

Define f(x) = | 3x - 9| + 8 .

Which of the following is an alternative way of defining ?

Possible Answers:

$f(x) = \left\{\begin{matrix}3x -17 & x \ge 3 \\ -3x+17 & x < 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} 3x+17 & x \ge 3 \\ - 3x -1 & x < 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix}3x -1 & x \ge 3 \\ -3x+17 & x < 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix}-3x+17 & x \ge 3 \\3x -1 & x < 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix}3x -1 & x \ge 3 \\ -3x+1 & x < 3 \end{matrix}\right.$

Correct answer:

$f(x) = \left\{\begin{matrix}3x -1 & x \ge 3 \\ -3x+17 & x < 3 \end{matrix}\right.$

Explanation:

If the expression between the absolute value bars is nonnegative, then | 3x - 9| = 3x-9 , and

f(x) = | 3x - 9| + 8

= 3x - 9 + 8

= 3x -1

This happens if:

$3x - 9 \ge 0$

$3x - 9+ 9 \ge 0 + 9$

$3x \ge 9$

$3x \div 3 \ge 9 \div 3$

$x \ge 3$

If the expression between the absolute value bars is negative, then | 3x - 9| = - (3x-9) = -3x+9 , and

f(x) = | 3x - 9| + 8

= -3x+9 + 8

= -3x+17

This happens if:

3x - 9 < 0 , or

x <3

Therefore, the function can be restated as

$f(x) = \left\{\begin{matrix}3x -1 & x \ge 3 \\ -3x+17 & x < 3 \end{matrix}\right.$ .

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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #134 : Algebraic Functions

Example Question #1061 : Algebra

Example Question #1061 : Algebra

Example Question #136 : Algebraic Functions

Example Question #137 : Algebraic Functions

Example Question #1061 : Algebra

Example Question #142 : Algebraic Functions

Example Question #143 : Algebraic Functions

Example Question #2702 : Act Math

Example Question #141 : Algebraic Functions

All SAT Math Resources

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