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PSAT Math : Algebra

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Example Questions

Example Question #32 : How To Find F(X)

Let f(x) be defined as follows:

$f(x)=\left\{\begin{matrix} 2x-3, &x<1 & \\ 4-x^2, &1\leq x< 3 & \\ |4x-9|, &x\geq 3 & \end{matrix}\right.$

What is the value of $f(1)+f(3)+f(5)?$

Possible Answers:

Correct answer:

Explanation:

The function f(x) is a piecewise function, which means that it is comprised of separate functions that change depending on the value of x. According to the problem, whenever x is less than 1, f(x) is defined by 2x - 3. Whenever x is greater than or equal to one and less than 3, f(x) is defined as $4-x^2$ . And whenever x is greater than or equal to 3, f(x) is equal to |4x – 9|.

The question asks us to find f(1) + f(3) + f(5). We need to calculate the values of f(1), f(3), and f(5) individually and then find their sum.

To find f(1), we must first decide which of the three possible functions for f(x) to use. Since f(1) means we are finding the value of f(x) when x = 1, we will have to use the second piece, which says that f(x) = $4-x^2$ . We can't use the function 2x - 3, because this is only valid when x < 1, not when x = 1.

$f(1)=4-(1)^2=4-1 = 3$

Next, we will find f(3). We need to use the third piece of the function which states that f(x) = |4x – 9|. Since f(3) means we are finding f(x) when x = 3, we can only use the third piece. The second piece, $4-x^2$ , is only defined if x is less than 3 and greater than or equal to 1.

$f(3)=|4x-9|=|4(3)-9|=|12-9|=|3|=3$

Lastly, we must find f(5). Again, we will use the function f(x) = |4x – 9|, because when x = 5, x must be greater than 3.

$f(5)=|4(5)-9|=|20-9|=|11|=11$

We can now add up f(1), f(3), and f(5), which would give us 3 + 3 + 11 = 17.

The answer is 17.

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Example Question #74 : Algebraic Functions

If a function, g, is defined by $\dpi{100} \small g\left ( x \right ) = \sqrt{2x+5}$ , what is $\dpi{100} \small \left [ g\left ( x \right )+2 \right ]^{2}$ when x = 2?

Possible Answers:

Correct answer:

Explanation:

First, let's expand $\dpi{100} \small \left [ g\left ( x \right )+2 \right ]^{2}$ . We will get the polynomial $\dpi{100} \small g\left ( x \right )^{2}+4g\left ( x \right )+4$ .

When we plug in the given function, g, into this polynomial and simplify, we will get $\dpi{100} \small 2x+5+4\sqrt{2x+5}+4$ .

Because this problem asks us for the value when x = 2, we have to plug in 2 for every x in the expression. $\dpi{100} \small 2\left ( 2 \right )+5+4\sqrt{2\left ( 2 \right)+5}+4= 25$ .

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Example Question #2781 : Sat Mathematics

In the xy-plane, a line with the equation $\dpi{100} \small y=\frac{1}{2}x-6$ crosses the y-axis at the point with the coordinates (m, n). What is the value of n?

Possible Answers:

$\dpi{100} \small \frac{1}{2}$

–6

Correct answer:

–6

Explanation:

This problem states that the given line crosses the y-axis at a certain point (m, n). By crossing the y-axis, we know that this point must be the y-intercept of this line. All y-intercepts lie on the y-axis and, therefore, have an x-coordinate of 0. If we plug in 0 for x in the equation of the line, $\dpi{100} \small y=\frac{1}{2}x-6$ , we will get

$\dpi{100} \small y=\frac{1}{2}\left (0 \right )-6$

y = –6

The y-intercept must be at the point

(0, –6)

so n must be –6.

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Example Question #2782 : Sat Mathematics

If $f(x)= x^3 + c$ and $\dpi{100} \small f(-3)=7$ . What is $\dpi{100} \small c$ ?

Possible Answers:

$31$

$7$

$30$

$-7$

$34$

Correct answer:

$34$

Explanation:

Plugging in $-3$ for $x$ and 7 for $f(x)$ into $f(x)= x^3 + c$ and solving for $c$ , we obtain $c=34$ .

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Example Question #2783 : Sat Mathematics

Solve for :

$3-z=3+z$ .

Possible Answers:

Correct answer:

Explanation:

Solving for yields

$2z=0\ or\ z=0$ .

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Example Question #2784 : Sat Mathematics

if $f(x) = x^{2} + x - 2$ and $g(x) = x - 3$ , solve $f(g(x))$ .

Possible Answers:

$f(g(x)) = x^{2} + 1$

$f(g(x)) = x^{2} + 2x - 5$

$f(g(x)) = x^{2} - 5x + 4$

$f(g(x)) = x^{2} + x - 3$

$f(g(x)) = x^{2} - 3$

Correct answer:

$f(g(x)) = x^{2} - 5x + 4$

Explanation:

We are solving for a composite function by substituting $g(x)$ into $f(x)$ to get: $f(g(x)) = (x - 3)^{2} + (x - 3) - 2$ before simplification.

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Example Question #2785 : Sat Mathematics

If $f(x)=2x^2-5$ where is an integer, which of the following could be a value of f(x) ?

II.

III.

Possible Answers:

II only

I only

I, II and III

I and III only

II and III only

Correct answer:

II and III only

Explanation:

Choice I is incorrect because to equal 0, $x^2=5$ , and since is an integer, this cannot be true.

Choice II is correct because f(x)=3 when x=2 or x=-2 .

Choice III is correct because f(x)=-3 when x=1 or x=-1 .

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Example Question #2786 : Sat Mathematics

Jamie is three times her little brother's age, and her little brother is two years younger than his older brother. Collectively, the three of them are 27 years old. How old is Jamie?

Possible Answers:

None of the available answers

Correct answer:

Explanation:

The algebraic expression for being Jamie's youngest brother's age is:

3x+x+(x+2)=27

5x+2=27

5x=25

x=5

Jamie's youngest brother is five, the next oldest brother is seven, and Jamie is 15.

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Example Question #961 : Psat Mathematics

Consider the function defined as follows:

$a \Delta b = (a-b)^3 +b$

Find:

$(3 \Delta 5) \times (2 \Delta 0)$

Possible Answers:

Correct answer:

Explanation:

The notation used above can be confusing. Let:

$a \Delta b = f(a,b)$

We can now find the answer by substituting the appropriate values into the equation:

$a \Delta b= f(a,b) = (a-b)^3 +b$

Therefore:

$3 \Delta 5 = f(3,5) = (3-5)^3 + 5 = (-2)^3 + 5 = -3$

$2 \Delta 0 = f(2,0) = (2-0)^3 + 0 = 8$

Finally:

$(3 \Delta 5) \times (2 \Delta 0) = (-3) \times 8 = -24$

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Example Question #2787 : Sat Mathematics

Solve for f(2) .

f(x)= 2x-22

Possible Answers:

$\frac{1}{18}$

$-\frac{1}{11}$

Correct answer:

Explanation:

To solve for f(2) , we actually have to solve for f(x) , when x=2 . We simply replace any $\small x$ with a $\small 2$ .

f(x) = 2x - 22

f(2)= 2(2)-22

f(2)= 4-22

f(2)= -18

The answer of f(x) when x=2 is .

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #32 : How To Find F(X)

Example Question #74 : Algebraic Functions

Example Question #2781 : Sat Mathematics

Example Question #2782 : Sat Mathematics

Example Question #2783 : Sat Mathematics

Example Question #2784 : Sat Mathematics

Example Question #2785 : Sat Mathematics

Example Question #2786 : Sat Mathematics

Example Question #961 : Psat Mathematics

Example Question #2787 : Sat Mathematics

All PSAT Math Resources

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