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We have to write an equation with an absolute value that will give the same answer for x=8 and x=14.
We can write a generic equation with a parameter and then find its value:
[tex]\begin{gathered} f(x)=|x-a| \\ f(8)=f(14) \end{gathered}[/tex]Then:
[tex]\begin{gathered} f(8)=|8-a| \\ f(14)=|14-a| \\ \Rightarrow|8-a|=|14-a| \end{gathered}[/tex]One of the two arguments, "8-a" or "14-a", has to be negative and the other positive. It must be "8-a" because if "14-a" were negative, then "8-a" would be also negative and there is no value of a that can make the two terms equal.
Then, we can rewrite the equality as:
[tex]-(8-a)=14-a[/tex]we then can solve for "a" as:
[tex]\begin{gathered} -(8-a)=14-a \\ a-8=14-a \\ a+a=14+8 \\ 2a=22 \\ a=\frac{22}{2} \\ a=11 \end{gathered}[/tex]Note that x=11 is the midpoint between x=8 and x=14.
Answer:
The absolute value function that gives the same value for x=8 and x=14 is f(x) = |x-11|
