QuestionFor $f(x)=7 x-6$ and $g(x)=\frac{1}{7}(x+6)$ , find $(f \circ g)(x)$ and $(g \circ f)(x)$ . Then determine whether $(f \circ g)(x)=(g \circ f)(x)$ .
What is $(f \circ g)(x)$ ? $(f \circ g)(x)=$

Studdy Solution

STEP 1

1. We have two functions: $f(x) = 7x - 6$ and $g(x) = \frac{1}{7}(x + 6)$ .
2. We need to find the composition of these functions, specifically $(f \circ g)(x)$ and $(g \circ f)(x)$ .
3. We will then check if $(f \circ g)(x) = (g \circ f)(x)$ .

STEP 2

1. Find $(f \circ g)(x)$ .
2. Find $(g \circ f)(x)$ .
3. Compare $(f \circ g)(x)$ and $(g \circ f)(x)$ .

STEP 3

To find $(f \circ g)(x)$ , substitute $g(x)$ into $f(x)$ :
$(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{7}(x + 6)\right)$

STEP 4

Substitute $\frac{1}{7}(x + 6)$ into $f(x) = 7x - 6$ :
$f\left(\frac{1}{7}(x + 6)\right) = 7\left(\frac{1}{7}(x + 6)\right) - 6$

STEP 5

Simplify the expression:
$7\left(\frac{1}{7}(x + 6)\right) = x + 6$
Thus:
$f\left(\frac{1}{7}(x + 6)\right) = x + 6 - 6$ $f\left(\frac{1}{7}(x + 6)\right) = x$
So, $(f \circ g)(x) = x$ .
The value of $(f \circ g)(x)$ is:
$(f \circ g)(x) = x$

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