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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #2811 : Sat Mathematics

If $f(t)=6+(4-t)^2$ , what is the smallest possible value of $f(t)$ ?

Possible Answers:

Correct answer:

Explanation:

This equation describes a parabola whose vertex is located at the point (4, 6). No matter how large or small the value of t gets, the smallest that f(t) can ever be is 6 because the parabola is concave up. To prove this to yourself you can plug in different values of t and see if you ever get anything smaller than 6.

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Example Question #2812 : Sat Mathematics

If $f(x)=x^{2}+3$ , then $f(x+h)=$ ?

Possible Answers:

$x^{2}+3+h$

$x^{2}+2xh+h^{2}$

$x^{2}+h^{2}+3$

$x^{2}+2xh+h^{2}+3$

$x^{2}+h^{2}$

Correct answer:

$x^{2}+2xh+h^{2}+3$

Explanation:

To find $f(x+h)$ when $f(x)=x^{2}+3$ , we substitute $(x+h)$ for $x$ in $f(x)$ .

Thus, $f(x+h)=(x+h)^{2}+3$ .

We expand $(x+h)^{2}$ to $x^{2}+xh+xh+h^{2}$ .

We can combine like terms to get $x^{2}+2xh+h^{2}$ .

We add 3 to this result to get our final answer.

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Example Question #2813 : Sat Mathematics

Let $f(x)$ and $g(x)$ be functions such that $f(x)=\frac{1}{x-3}$ , and $f(g(x))=g(f(x))=x$ . Which of the following is equal to $g(x)$ ?

Possible Answers:

$\frac{4}{x}$

$\frac{1}{x+3}$

$\frac{x}{x-3}$

$\frac{3x+1}{x}$

$\frac{x}{x+3}$

Correct answer:

$\frac{3x+1}{x}$

Explanation:

If $\dpi{100} h(x)$ and $\dpi{100} k(x)$ are defined as inverse functions, then $\dpi{100} h(k(x))=k(h(x))=x$ . Thus, according to the definition of inverse functions, $\dpi{100} f(x)$ and $\dpi{100} g(x)$ given in the problem must be inverse functions.

If we want to find the inverse of a function, the most straighforward method is usually replacing $\dpi{100} f(x)$ with $\dpi{100} y$ , swapping $\dpi{100} y$ and $\dpi{100} x$ , and then solving for $\dpi{100} y$ .

We want to find the inverse of $f(x)=\frac{1}{x-3}$ . First, we will replace $\dpi{100} f(x)$ with $\dpi{100} y$ .

$y = \frac{1}{x-3}$

Next, we will swap $\dpi{100} x$ and $\dpi{100} y$ .

$x = \frac{1}{y-3}$

Lastly, we will solve for $\dpi{100} y$ . The equation that we obtain in terms of $\dpi{100} x$ will be in the inverse of $\dpi{100} f(x)$ , which equals $\dpi{100} g(x)$ .

We can treat $x = \frac{1}{y-3}$ as a proportion, $\frac{x}{1} = \frac{1}{y-3}$ . This allows us to cross multiply and set the results equal to one another.

$x(y-3)= 1$

We want to get y by itself, so let's divide both sides by x.

$y-3=\frac{1}{x}$

Next, we will add 3 to both sides.

$y=\frac{1}{x}+3$

To combine the right side, we will need to rewrite 3 so that it has a denominator of $\dpi{100} x$ .

$y=\frac{1}{x}+3 = \frac{1}{x}+\frac{3x}{x}=\frac{3x+1}{x}$

The answer is $\frac{3x+1}{x}$ .

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Example Question #987 : Psat Mathematics

$\small \textup{The function}\, f(x)\, \textup{is defined as }f\left ( x\right )=5x-2$ .

$\small \small f\left ( 4\right )-f\left ( 2\right )=\,?$

Possible Answers:

$\small 14$

$\small 8$

$\small 2$

$\small 6$

$\small 10$

Correct answer:

$\small 10$

Explanation:

$\small f\left ( x\right )=5x - 2$

$\small f\left ( 4\right )=5\left ( 4\right )-2 = 18$

$\small f\left ( 2\right )=5\left ( 2\right )-2 = 8$

$\small 18-8 = 10$

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Example Question #2814 : Sat Mathematics

Let the function f be defined by f(x)=x-t. If f(12)=4, what is the value of f(0.5*t)?

Possible Answers:

Correct answer:

Explanation:

First we substitute in 12 for x and set the equation up as 12-t=4. We then get t=8, and substitute that for t and get f(0.5*8), giving us f(4). Plugging 4 in for x, and using t=8 that we found before, gives us:

f(4) = 4 - 8 = -4

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Example Question #332 : Algebra

What is the value of the function f(x) = 6x²+ 16x – 6 when x = –3?

Possible Answers:

–108

–12

Correct answer:

Explanation:

There are two ways to do this problem. The first way just involves plugging in –3 for x and solving 6〖(–3)〗²+ 16(–3) – 6, which equals 54 – 48 – 6 = 0. The second way involves factoring the polynomial to (6x – 2)(x + 3) and then plugging in –3 for x. The second way quickly shows that the answer is 0 due to multiplying by (–3 + 3).

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Example Question #2815 : Sat Mathematics

Given the functions f(x) = 2x + 4 and g(x) = 3x – 6, what is f(g(x)) when x = 6?

Possible Answers:

192

144

Correct answer:

Explanation:

We need to work from the inside to the outside, so g(6) = 3(6) – 6 = 12.

Then f(g(6)) = 2(12) + 4 = 28.

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Example Question #36 : Algebraic Functions

A function f(x) = –1 for all values of x. Another function g(x) = 3x for all values of x. What is g(f(x)) when x = 4?

Possible Answers:

–1

–3

–12

Correct answer:

–3

Explanation:

We work from the inside out, so we start with the function f(x). f(4) = –1. Then we plug that value into g(x), so g(f(x)) = 3 * (–1) = –3.

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Example Question #37 : Algebraic Functions

What is f(–3) if f(x) = x² + 5?

Possible Answers:

–14

–4

Correct answer:

Explanation:

f(–3) = (–3)² + 5 = 9 + 5 = 14

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Example Question #2816 : Sat Mathematics

For all values of x, f(x) = 7x² – 3, and for all values of y, g(y) = 2y + 9. What is g(f(x))?

Possible Answers:

14x² – 3

14y² + 3

7y² – 3

14x² + 3

2x + 9

Correct answer:

14x² + 3

Explanation:

The inner function f(x) is like our y-value that we plug into g(y).

g(f(x)) = 2(7x² – 3) + 9 = 14x² – 6 + 9 = 14x² + 3.

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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #2811 : Sat Mathematics

Example Question #2812 : Sat Mathematics

Example Question #2813 : Sat Mathematics

Example Question #987 : Psat Mathematics

Example Question #2814 : Sat Mathematics

Example Question #332 : Algebra

Example Question #2815 : Sat Mathematics

Example Question #36 : Algebraic Functions

Example Question #37 : Algebraic Functions

Example Question #2816 : Sat Mathematics

All SAT Math Resources

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