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Precalculus : Special Functions

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Example Questions

Example Question #1 : Piecewise Functions

Let

$y=\left\{\begin{matrix} 2x&x\leq 0\\ x+9& x>0 \end{matrix}\right.$ $\displaystyle y=\left\{\begin{matrix} 2x&x\leq 0\\ x+9& x>0 \end{matrix}\right.$

What does $\displaystyle y$ equal when x=3 $\displaystyle x=3$ ?

Possible Answers:

$\displaystyle 10$

$\displaystyle 6$

$\displaystyle 3$

$\displaystyle 12$

Correct answer:

$\displaystyle 12$

Explanation:

Because 3>0 we plug the x value into the bottom equation.

x=3 $\displaystyle x=3$

y=x+9 $\displaystyle y=x+9$

$\displaystyle y=3+9=12$

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Example Question #1 : Special Functions

Let

$y=\left\{\begin{matrix} x-9&x\leq 0\\ x+9& x>0 \end{matrix}\right.$ $\displaystyle y=\left\{\begin{matrix} x-9&x\leq 0\\ x+9& x>0 \end{matrix}\right.$

What does $\displaystyle y$ equal when x=0 $\displaystyle x=0$ ?

Possible Answers:

$\displaystyle 0$

$\displaystyle 8$

$\displaystyle -9$

$\displaystyle 9$

Correct answer:

$\displaystyle -9$

Explanation:

Because $0\leq0$ $\displaystyle 0\leq0$ we use the first equation.

y=x-9 $\displaystyle y=x-9$

Therefore, plugging in x=0 into the above equation we get the following,

$\displaystyle y=0-9=-9$ .

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Example Question #1 : Piecewise Functions

Determine the value of f(1) $\displaystyle f(1)$ if the function is

$f(x)=\left\{\begin{matrix} x& x<3\\ \pi&x=3 \\ 4&x>3 \end{matrix}\right.$ $\displaystyle f(x)=\left\{\begin{matrix} x& x< 3\\ \pi&x=3 \\ 4&x>3 \end{matrix}\right.$

Possible Answers:

f(1)=1 $\displaystyle f(1)=1$

f(1)=3 $\displaystyle f(1)=3$

f(1)=4 $\displaystyle f(1)=4$

$f(1)=\pi$ $\displaystyle f(1)=\pi$

Correct answer:

f(1)=1 $\displaystyle f(1)=1$

Explanation:

In order to determine the value of f(1) $\displaystyle f(1)$ of the function we set x=1. $\displaystyle x=1.$

The value comes from the function in the first row of the piecewise function, and as such

f(1)=1 $\displaystyle f(1)=1$

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Example Question #1 : Piecewise Functions

Determine the value of f(1) $\displaystyle f(1)$ if the function is

$f(x)=\left\{\begin{matrix} \sqrt{x}& 0\leq x<4\\ \3&x=4 \\ e^{-x}&x>4 \end{matrix}\right.$

Possible Answers:

$f(1)=\frac{1}{e}$ $\displaystyle f(1)=\frac{1}{e}$

f(1)=3 $\displaystyle f(1)=3$

f(1)=1 $\displaystyle f(1)=1$

f(1)=3 $\displaystyle f(1)=3$

Correct answer:

f(1)=1 $\displaystyle f(1)=1$

Explanation:

In order to determine the value of f(1) $\displaystyle f(1)$ of the function we set x=1. $\displaystyle x=1.$

The value comes from the function in the first row of the piecewise function, and as such

$f(1)=\sqrt{1}=1$ $\displaystyle f(1)=\sqrt{1}=1$

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Example Question #2 : Special Functions

For the function f(x) $\displaystyle f(x)$ defined below, what is the value of f(x) $\displaystyle f(x)$ when x=3 $\displaystyle x=3$ ?

$f(x)=\left\{\begin{matrix} -x+1&\text{ for }x<1\\ x^{2}-2&\text{ for }1< x< 3\\ 2^{x}&\text{ for }x\geq 3 \end{matrix}$

Possible Answers:

-2

Correct answer:

Explanation:

Evaluate the function for x=3 $\displaystyle x=3$ . Based on the domains of the three given expressions, you would use $2^{x}$ $\displaystyle 2^{x}$ , since $\displaystyle x$ is greater than or equal to $\displaystyle 3$ .

$2^{x}=2^{3}=8$ $\displaystyle 2^{x}=2^{3}=8$

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Example Question #1 : Special Functions

If $\displaystyle F$ is the greatest integer function, what is the value of F(2.4) $\displaystyle F(2.4)$ ?

Possible Answers:

$\displaystyle 1$

$\displaystyle 4$

2.5 $\displaystyle 2.5$

$\displaystyle 2$

$\displaystyle 3$

Correct answer:

$\displaystyle 2$

Explanation:

The greatest integer function takes an input and produces the greatest integer less than the input. Thus, the output is always smaller than the input and is an integer itself. Since our input was 2.4 $\displaystyle 2.4$ , we are looking for an integer less than this, which must be $\displaystyle 2$ since any smaller integer would by definition not be "greatest".

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Example Question #1 : Absolute Value Functions

Which of the following is a point on the following function?

$y=\left | x^2-5x-56\right |+27$ $\displaystyle y=\left | x^2-5x-56\right |+27$

Possible Answers:

(0,0) $\displaystyle (0,0)$

$\displaystyle (-27,-27)$

(5,83) $\displaystyle (5,83)$

(5,-43) $\displaystyle (5,-43)$

(14,36) $\displaystyle (14,36)$

Correct answer:

(5,83) $\displaystyle (5,83)$

Explanation:

One way to approach this problem would be to plug in each answer and see what works. However, I would be a little more strategic and eliminate any options that don't make sense.

Our y value will never be negative, so eliminate any options with a negative y-value.

Try (0,0) really quick, since it's really easy

The only point that makes sense is (5,83), therefore it is the correct answer

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Example Question #31 : Functions

Evaluate: $\left | x-3\right |<15$ $\displaystyle \left | x-3\right |< 15$

Possible Answers:

x<18 $\displaystyle x< 18$

$-12\leq x\leq18$ $\displaystyle -12\leq x\leq18$

-12<x<18 $\displaystyle -12< x< 18$

$x\leq18$ $\displaystyle x\leq18$

-12,18 $\displaystyle -12,18$

Correct answer:

-12<x<18 $\displaystyle -12< x< 18$

Explanation:

Cancel the absolute value sign by separating the function $\left | x-3\right |<15$ $\displaystyle \left | x-3\right |< 15$ into its positive and negative counterparts.

x-3<15 $\displaystyle x-3< 15$

-(x-3)<15 $\displaystyle -(x-3)< 15$

Evaluate the first scenario.

x-3<15 $\displaystyle x-3< 15$

x<18 $\displaystyle x< 18$

Evaluate the second scenario.

-(x-3)<15 $\displaystyle -(x-3)< 15$

x-3>-15 $\displaystyle x-3>-15$

x>-12 $\displaystyle x>-12$

The correct answer is:

-12<x<18 $\displaystyle -12< x< 18$

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Example Question #2 : Absolute Value Functions

If $y=\left |x^3-5x^2+5 \right |$ $\displaystyle y=\left |x^3-5x^2+5 \right |$ , then what is the value of $\displaystyle y$ when x=3 $\displaystyle x=3$ ?

Possible Answers:

-7

-13

Correct answer:

Explanation:

We evaluate for x=3 $\displaystyle x=3$

$\displaystyle y=|(3)^3-5(3)^2+5|$

$\displaystyle y=|27-45+5|$

y=|-13| $\displaystyle y=|-13|$

Since the absolute value of any number represents its magnitude from $\displaystyle 0$ and is therefore always positive, the final answer would be y=13 $\displaystyle y=13$

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Precalculus : Special Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Piecewise Functions

Example Question #1 : Special Functions

Example Question #1 : Piecewise Functions

Example Question #1 : Piecewise Functions

Example Question #2 : Special Functions

Example Question #1 : Special Functions

Example Question #1 : Absolute Value Functions

Example Question #31 : Functions

Example Question #2 : Absolute Value Functions

All Precalculus Resources

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