Precalculus : Absolute Value Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Special Functions

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

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

 

Possible Answers:

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

\(\displaystyle (5,83)\)

\(\displaystyle (0,0)\)

\(\displaystyle (-27,-27)\)

\(\displaystyle (14,36)\)

Correct answer:

\(\displaystyle (5,83)\)

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

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

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

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

Example Question #2 : Special Functions

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

Possible Answers:

\(\displaystyle -12,18\)

\(\displaystyle x< 18\)

\(\displaystyle x\leq18\)

\(\displaystyle -12< x< 18\)

\(\displaystyle -12\leq x\leq18\)

Correct answer:

\(\displaystyle -12< x< 18\)

Explanation:

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

\(\displaystyle x-3< 15\)

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

Evaluate the first scenario.

\(\displaystyle x-3< 15\)

\(\displaystyle x< 18\)

Evaluate the second scenario.

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

\(\displaystyle x-3>-15\)

\(\displaystyle x>-12\)

The correct answer is:

\(\displaystyle -12< x< 18\)

Example Question #2 : Special Functions

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

Possible Answers:

-7

-13

20

7

13

Correct answer:

13

Explanation:

We evaluate for \(\displaystyle x=3\)

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

\(\displaystyle y=|27-45+5|\)

\(\displaystyle y=|-13|\)

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

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