Chapter 3.8, Problem 57HP

To determine

To determine whether the graph of an absolute value function can have more than one $x$ -intercepts or not.

Answer to Problem 57HP

Tayden is correct.

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The zero of the function is the $x$ -intercept of that function.

Calculation:

Consider the general form of an absolute value function:

  $f(x)=a\left|x-h\right|+k$

To find the $x$ -intercepts of $f(x)$ , put $f(x)=0$

  $\begin{array}{l}\Rightarrow 0=a\left|x-h\right|+k\\ \Rightarrow -k=a\left|x-h\right|\\ \Rightarrow -\frac{k}{a}=\left|x-h\right|\\ \Rightarrow \left(x-h\right)=\pm \left(-\frac{k}{a}\right)\\ \Rightarrow \left(x-h\right)=\mp \left(\frac{k}{a}\right)\\ \Rightarrow x=\mp \left(\frac{k}{a}\right)+h\end{array}$

Therefore, there are two $x$ -intercepts: $\left(-\frac{k}{a}+h,0\right)$ and $\left(\frac{k}{a}+h,0\right)$ .

Conclusion:

Therefore, the absolute value function can have two $x$ -intercepts.

Therefore, Tayden is correct.