College

Functions [tex]f(x)[/tex] and [tex]g(x)[/tex] are defined below:

[tex]\begin{array}{l} f(x)=|x-5|-6 \\ g(x)=4|x-5|-9 \end{array}[/tex]

Determine where [tex]f(x) = g(x)[/tex] by constructing a table of values.

A. [tex]x=-4[/tex]
B. [tex]x=-6[/tex]
C. [tex]x=6[/tex]
D. [tex]x=4 \text{ and } 6[/tex]

Answer :

We are given the functions
[tex]$$
f(x)=|x-5|-6 \quad\text{and}\quad g(x)=4|x-5|-9.
$$[/tex]

To find where [tex]$f(x)=g(x)$[/tex], we set the two functions equal:

[tex]$$
|x-5|-6=4|x-5|-9.
$$[/tex]

### Step 1. Isolate the Absolute Value Term

First, add 9 to both sides:

[tex]$$
|x-5|-6+9=4|x-5|.
$$[/tex]

This simplifies to

[tex]$$
|x-5|+3=4|x-5|.
$$[/tex]

Next, subtract [tex]$|x-5|$[/tex] from both sides:

[tex]$$
3=3|x-5|.
$$[/tex]

### Step 2. Solve for the Absolute Value

Divide both sides by 3:

[tex]$$
|x-5|=1.
$$[/tex]

The equation [tex]$|x-5|=1$[/tex] means the expression inside the absolute value can be either [tex]$1$[/tex] or [tex]$-1$[/tex]. This gives us two equations:

1. [tex]$$x-5=1,$$[/tex]
2. [tex]$$x-5=-1.$$[/tex]

### Step 3. Solve Each Equation

For the first equation:
[tex]$$
x-5=1 \quad \Longrightarrow \quad x=6.
$$[/tex]

For the second equation:
[tex]$$
x-5=-1 \quad \Longrightarrow \quad x=4.
$$[/tex]

Thus, the two points where [tex]$f(x)=g(x)$[/tex] are at [tex]$x=4$[/tex] and [tex]$x=6$[/tex].

### Step 4. Verify by Constructing a Table of Values

We can verify these results by constructing a table of values for selected [tex]$x$[/tex] values:

[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x)=|x-5|-6 & g(x)=4|x-5|-9 \\
\hline
0 & |0-5|-6=|{-5}|-6=5-6=-1 & 4\cdot|0-5|-9=4\cdot5-9=20-9=11 \\
1 & |1-5|-6=|{-4}|-6=4-6=-2 & 4\cdot|1-5|-9=4\cdot4-9=16-9=7 \\
2 & |2-5|-6=|{-3}|-6=3-6=-3 & 4\cdot|2-5|-9=4\cdot3-9=12-9=3 \\
3 & |3-5|-6=|{-2}|-6=2-6=-4 & 4\cdot|3-5|-9=4\cdot2-9=8-9=-1 \\
4 & |4-5|-6=|{-1}|-6=1-6=-5 & 4\cdot|4-5|-9=4\cdot1-9=4-9=-5 \\
5 & |5-5|-6=0-6=-6 & 4\cdot|5-5|-9=4\cdot0-9=0-9=-9 \\
6 & |6-5|-6=1-6=-5 & 4\cdot|6-5|-9=4\cdot1-9=4-9=-5 \\
7 & |7-5|-6=2-6=-4 & 4\cdot|7-5|-9=4\cdot2-9=8-9=-1 \\
8 & |8-5|-6=3-6=-3 & 4\cdot|8-5|-9=4\cdot3-9=12-9=3 \\
9 & |9-5|-6=4-6=-2 & 4\cdot|9-5|-9=4\cdot4-9=16-9=7 \\
10 & |10-5|-6=5-6=-1 & 4\cdot|10-5|-9=4\cdot5-9=20-9=11 \\
\hline
\end{array}
\][/tex]

In this table, you can see that at [tex]$x=4$[/tex] (with [tex]$f(4)=-5$[/tex] and [tex]$g(4)=-5$[/tex]) and at [tex]$x=6$[/tex] (with [tex]$f(6)=-5$[/tex] and [tex]$g(6)=-5$[/tex]) the two functions are equal.

### Final Answer

The functions [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] are equal when [tex]$x=4$[/tex] and [tex]$x=6$[/tex].