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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #17 : Functions

Finding an inverse function.

Given

f(x)=2x+1

find

f^-1(x) .

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Finding an inverse function is essentially finding the output in terms of the input. To do this switch the independent and dependent variables. First set

f(x)=y .

Then

y=2x+1

Now, to switch the role of the variables solve for x in terms of y.

y-1 = 2x

2x=y-1

$x=\frac{1}2{}y-\frac{1}{2}$

So, y is now the independent variable, and x is the dependent variable. The final step is to write the inverse function in proper notation.

Set $x=f^{-1}(x)$ , and y=x .

This gives

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

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Example Question #12 : Functions

Which function below correctly translates f(x) four units to the right and two units down?

Possible Answers:

f(x-4)+2

f(x+4)-2

f(x+4)+2

f(x-4)-2

Correct answer:

f(x-4)-2

Explanation:

The translation f(x+h) moves the graph of f(x) units to the left if h>0 . To see this, compare the graphs of y=x and y=x+1 . A translation to the right must be in the form f(x-h) if h>0 .

A vertical translation is up if k>0 . We want to move the function down, so our translation subtracts two from f(x-h) .

Combining these, we get f(x-4)-2 .

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Example Question #21 : Relations And Functions

What is the domain of the following function?

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

Possible Answers:

$\text{The set of all reals}$

$\mathbb{R}-\{{(2k+1)\pi\, ,$ k\in 0,1,2,3,4,.......\}$

$\mathbb{R}-\{\pi\}$

- $\mathbb{R}-\{\pi ,3\pi\}$

$\mathbb{R}-\{-\pi\}$

Correct answer:

$\mathbb{R}-\{{(2k+1)\pi\, ,$ k\in 0,1,2,3,4,.......\}$

Explanation:

The denominator becomes zero whenever cos(x) becomes -1 and this happens when x is an odd multiple of $\pi$ . To avoid division by zero, we exclude all these numbers. Therefore the domain is:

$\mathbb{R}-\{{(2k+1)\pi\, ,$ k\in 0,1,2,3,4,.......\}$

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Example Question #4 : Angles

Solve for .

(Figure not drawn to scale).

Possible Answers:

10^o

12^o

8^o

14^o

Correct answer:

14^o

Explanation:

The angles are supplementary, therefore, the sum of the angles must equal 180^o .

$\small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$

$\small 12n=168^o$

$\small n=14^o$

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Example Question #6 : Understanding Complementary And Suplmentary Angles

Are $129^{\circ}$ and $51^{\circ}$ supplementary angles?

Possible Answers:

Yes

Not enough information

Correct answer:

Yes

Explanation:

Since supplementary angles must add up to $180^{\circ}$ , the given angles are indeed supplementary.

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Example Question #1 : Angles

Solve for and .

(Figure not drawn to scale).

Possible Answers:

$\small x=15^o;\ y=7.5^o$

$\small x=15^o;\ y=37.5^o$

$\small x=7.5^o;\ y=18.75^o$

$\small x=15^o;\ y=52.5^o$

Correct answer:

$\small x=15^o;\ y=37.5^o$

Explanation:

The angles containing the variable all reside along one line, therefore, their sum must be 180^o .

$\small 5x+4x+3x=180^o$

$\small 12x=180^o$

$\small x=15^o$

Because and are opposite angles, they must be equal.

$\small 2y=5x$

$\small x=15^o$

$\small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$

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Example Question #52 : Pre Calculus

For what values of will the given polynomial pass through the x-axis if plotted in Cartesian coordinates?

f(x)=(x+2)^2(x-1)(x+3)(x-5)^3

Possible Answers:

x= -3,-2,1,5

x= -3,1

None of the other answers

x=-3,1,5

x= -2,5

Correct answer:

x=-3,1,5

Explanation:

One can remember that if the multiplicity of a zero is odd then it passes through the x-axis and if it's even then it 'bounces' off the x-axis. You can think about this analytically as well. What happens when we plug a number into our function just slightly above or below a zero with an even multiplicity? You find that the sign is always positive. Whereas a zero with an odd multiplicity will yield a positive on one side and a negative on the other. For zeros with odd multiplicity this alters the sign of our output and the function passes through the x-axis. Whereas the zero with even multiplicity will output a number with the same sign just above and below its zero, thus it 'bounces' off the x-axis.

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Example Question #53 : Pre Calculus

For this particular question we are restricting the domain of both $f \, \textup{and}\,g$ to nonnegative values, or the interval $\left [0, \right \infty )$ .

Let f(x)=x^2+4 and g(x) = x^3 .

For what values of is f(x) > g(x) ?

Possible Answers:

$[2,\infty]$

$[4,\infty)$

[0,8)

[0,2)

Correct answer:

[0,2)

Explanation:

The cubic function will increase more quickly than the quadratic, so the quadratic function must have a head start. At x=2 , both functions evaluate to 8. After than point, the cubic function will increase more quickly.

The domain was restricted to nonnegative values, so this interval is our only answer.

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Example Question #1 : Graphing The Sine And Cosine Functions

Triangle

What is the $\sin C$ ?

Possible Answers:

$\frac{BC}{AB}$

$\frac{BC}{AC}$

$\frac{AC}{AB}$

$\frac{AB}{AC}$

$\frac{AB}{BC}$

Correct answer:

$\frac{AB}{AC}$

Explanation:

$\sin X = \frac{OPPOSITE}{HYPOTENUSE}$

$\sin C = \frac{AB}{AC}$

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Example Question #2 : Graphing The Sine And Cosine Functions

Triangle

In the right triangle above, which of the following expressions gives the length of y?

Possible Answers:

$x\cos\theta$

$x\tan\theta$

$\frac{\sin(x)}{\theta}$

$\cos(x\times \theta)$

$x\sin\theta$

Correct answer:

$x\cos\theta$

Explanation:

$\cos\theta$ is defined as the ratio of the adjacent side to the hypotenuse, or in this case $\frac{y}{x}$ . Solving for y gives the correct expression.

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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #17 : Functions

Example Question #12 : Functions

Example Question #21 : Relations And Functions

Example Question #4 : Angles

Example Question #6 : Understanding Complementary And Suplmentary Angles

Example Question #1 : Angles

Example Question #52 : Pre Calculus

Example Question #53 : Pre Calculus

Example Question #1 : Graphing The Sine And Cosine Functions

Example Question #2 : Graphing The Sine And Cosine Functions

All Precalculus Resources

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