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High School Math : Functions and Graphs

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Example Questions

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Example Question #1 : Understanding Functional Notations

If $f(x)=3x^{2}+5$ and g(x)= x-2 , what is g(f(x)) ?

Possible Answers:

$x^{2}-2$

$3x^{2}+3$

$3x^{2}-12x+17$

$2x^{2}+5$

$3x^{2}+12x-7$

Correct answer:

$3x^{2}+3$

Explanation:

g(f(x)) means f(x) gets plugged into g(x) .

Thus $g(f(x))= (3x^{2}+5) - 2 = 3x^{2}+3$ .

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Example Question #3 : Understanding Functional Notations

Let $f(x)= x^{3}-2x^{2}+x$ and $g(x)=x^{2}-1$ . What is f(g(2)) ?

Possible Answers:

Correct answer:

Explanation:

Calculate g(2) and plug it into f(x) .

$g(2)= (2)^{2}-1=4-1=3$

$f(3)=(3)^{3}-2(3)^{2}+(3)=27-2(9)+(3)=27-18+3=12$

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Example Question #1 : Function Notation

Evaluate $\small f(g(3))$ if f(x)=6x-4 and g(x)=x^2 .

Possible Answers:

Undefined

196

Correct answer:

Explanation:

$\small f(g(3))$

This expression is the same as saying "take the answer of $\small g(3)$ and plug it into $\small f(x)$ ."

First, we need to find $\small g(3)$ . We do this by plugging $\small 3$ in for $\small x$ in $\small g(x)$ .

g(x)=x^2

g(3)=3^2=9

Now we take this answer and plug it into $\small f(x)$ .

f(g(3))=f(9)

We can find the value of $\small f(9)$ by replacing $\small x$ with $\small 9$ .

f(x)=6x-4

f(9)=6(9)-4=50

This is our final answer.

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Example Question #1 : Transformations Of Parabolic Functions

If the function f(x) is depicted here, which answer choice graphs f(x+2) ?

Possible Answers:

None of these graphs are correct.

Correct answer:

Explanation:

The function f(x+b) shifts a function f(x) units to the left. Conversely, f(x-b) shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function f(x)+k or f(x)-k .

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Example Question #1 : Understanding Inverse Functions

Let f(x)=5-(1+x)^3 . What is $f^{-1}(x)$ ?

Possible Answers:

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

$f^{-1}(x)=\left (1-x)\right ^{1/3}+5$

$f^{-1}(x)=\left ((5-x)^{3}-1 )\right ^{1/3}$

$f^{-1}(x)=(5-x)^{3}-1$

$f^{-1}(x)=\left (1-5x)\right ^{1/3}+1$

Correct answer:

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

Explanation:

We are asked to find $f^{^{-1}} (x)$ , which is the inverse of a function.

In order to find the inverse, the first thing we want to do is replace f(x) with y. (This usually makes it easier to separate x from its function.).

y=5-(1+x)^3

Next, we will swap x and y.

x=5-(1+y)^3

Then, we will solve for y. The expression that we determine will be equal to $f^{^{-1}} (x)$ .

x=5-(1+y)^3

Subtract 5 from both sides.

x-5=-(1+y)^3

Multiply both sides by -1.

-1(x-5)=5-x=(1+y)^3

We need to raise both sides of the equation to the 1/3 power in order to remove the exponent on the right side.

$(5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}$

We will apply the general property of exponents which states that $(a^b)^{c}=a^{b\cdot c}$ .

$(5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}=(1+y)^{3\cdot \frac{1}{3}}=(1+y)^1=1+y$

Laslty, we will subtract one from both sides.

$(5-x)^{\frac{1}{3}}-1=y$

The expression equal to y is equal to the inverse of the original function f(x). Thus, we can replace y with $f^{-1}(x)$ .

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

The answer is $f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ .

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Example Question #2 : Understanding Inverse Functions

What is the inverse of $y=4x^{2}$ ?

Possible Answers:

$y=-4x^{2}$

$y=\pm \sqrt{x}$

$y=\frac{\pm \sqrt{x}}{4}$

$y=\frac{\pm \sqrt{x}}{2}$

$y=\frac{x^{2}}{4}$

Correct answer:

$y=\frac{\pm \sqrt{x}}{2}$

Explanation:

The inverse of requires us to interchange and and then solve for .

$y=4x^{2}$

$x=4y^{2}$

Then solve for :

$y=\frac{\pm \sqrt{x}}{2}$

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Example Question #3 : Understanding Inverse Functions

If $f(x)=x^{2}+1$ , what is $f^{-1}(x)$ ?

Possible Answers:

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

$f^{-1}(x)=\sqrt{x-1}$

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

$f^{-1}(x)=x^{2}-1$

$f^{-1}(x)=\sqrt{x^{2}+1}$

Correct answer:

$f^{-1}(x)=\sqrt{x-1}$

Explanation:

To find the inverse of a function, exchange the and variables and then solve for .

$x=y^{2}+1$

$x-1=y^{2}$

$\sqrt{x-1}=y=f^{-1}(x)$

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Example Question #1 : Vertical And Horizontal Lines

Which of the following is a horizontal line?

Possible Answers:

y=2x+1

y = 0

x = 4

y = 3x + 4

y = -x

Correct answer:

y = 0

Explanation:

A horizontal line has infinitely many values for , but only one possible value for . Thus, it is always of the form y = c , where is a constant. Horizontal lines have a slope of . The only equation of this form is y = 0 .

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Example Question #1 : Understanding Vertical And Horizontal Lines

Which of the following is a vertical line?

Possible Answers:

x = 3

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

y = 3x + 2

y = 4

y = x

Correct answer:

x = 3

Explanation:

A vertical line is one in which the values can vary. Namely, there is only one possible value for , and can be any number. Thus, by this description, the only vertical line listed is x = 3 .

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Example Question #1 : Linear Functions

Which of the following has a slope of 0?

Possible Answers:

$y = 3^{x}$

x = 0

y = 10

y = x

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

Correct answer:

y = 10

Explanation:

A line with a slope of zero will be horizontal. A horizontal line has only one possible value for , and can be any value.

Thus, the only given equation which fits this description is y = 10 .

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High School Math : Functions and Graphs

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Functional Notations

Example Question #3 : Understanding Functional Notations

Example Question #1 : Function Notation

Example Question #1 : Transformations Of Parabolic Functions

Example Question #1 : Understanding Inverse Functions

Example Question #2 : Understanding Inverse Functions

Example Question #3 : Understanding Inverse Functions

Example Question #1 : Vertical And Horizontal Lines

Example Question #1 : Understanding Vertical And Horizontal Lines

Example Question #1 : Linear Functions

All High School Math Resources

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