Special Functions - Pre-Calculus

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Question

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

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

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Answer

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

$y=\left | x^2-5x-56\right |+27 \rightarrow 0=\left|0^2-5(0)-56\right|+27=83 \rightarrow 0\neq 83$

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

$y=\left | x^2-5x-56\right |+27 \rightarrow 83=\left|5^2-5(5)-56\right|+27=83 \rightarrow 83=83$

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Question

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

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Answer

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

Evaluate the first scenario.

Evaluate the second scenario.

The correct answer is:

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Question

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

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Answer

We evaluate for

Since the absolute value of any number represents its magnitude from and is therefore always positive, the final answer would be

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Question

If is the greatest integer function, what is the value of ?

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Answer

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 , we are looking for an integer less than this, which must be since any smaller integer would by definition not be "greatest".

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Question

For the function defined below, what is the value of when ?

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

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Answer

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

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

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Question

For the function defined below, what is the value of when ?

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

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Answer

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

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

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Question

Let

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

What does equal when ?

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Answer

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

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Question

Let

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

What does equal when ?

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Answer

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

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Question

Let

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

What does equal when ?

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Answer

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

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

.

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Question

Determine the value of if the function is

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

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Answer

In order to determine the value of of the function we set

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

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Question

Determine the value of if the function is

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

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Answer

In order to determine the value of of the function we set

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

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

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Question

For the function defined below, what is the value of when ?

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

Tap to reveal answer

Answer

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

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

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Question

Let

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

What does equal when ?

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Answer

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

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Question

Let

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

What does equal when ?

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Answer

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

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

.

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Question

Determine the value of if the function is

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

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Answer

In order to determine the value of of the function we set

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

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Question

Determine the value of if the function is

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

Tap to reveal answer

Answer

In order to determine the value of of the function we set

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

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

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Question

For the function defined below, what is the value of when ?

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

Tap to reveal answer

Answer

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

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

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Question

Let

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

What does equal when ?

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Answer

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

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

.

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Question

Determine the value of if the function is

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

Tap to reveal answer

Answer

In order to determine the value of of the function we set

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

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Question

Determine the value of if the function is

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

Tap to reveal answer

Answer

In order to determine the value of of the function we set

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

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

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