QuestionCheck if $(f \circ g)(x)$ equals $(g \circ f)(x)$ for $f(x)=7x-2$ and $g(x)=2x-7$ . Simplify $(f \circ g)(x)=\square($ $)$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $7x -$ . The function $g(x)$ is defined as $x -7$
3. We need to check if the composition of functions $(f \circ g)(x)$ is equivalent to $(g \circ f)(x)$

STEP 2

First, let's find $(f \circ g)(x)$ , which means we apply the function $g(x)$ first and then apply the function $f(x)$ to the result. $(f \circ g)(x) = f(g(x))$

STEP 3

Now, substitute $g(x)$ into $f(x)$ .
$f(g(x)) =7g(x) -2$

STEP 4

Substitute the definition of $g(x)$ into the equation.
$f(g(x)) =7(2x -7) -2$

STEP 5

implify the equation.
$f(g(x)) =14x -49 -2$

STEP 6

Further simplify the equation.
$f(g(x)) =14x -51$

STEP 7

Now, let's find $(g \circ f)(x)$ , which means we apply the function $f(x)$ first and then apply the function $g(x)$ to the result.
$(g \circ f)(x) = g(f(x))$

STEP 8

Now, substitute $f(x)$ into $g(x)$ .
$g(f(x)) =2f(x) -7$

STEP 9

Substitute the definition of $f(x)$ into the equation.
$g(f(x)) =2(7x -2) -7$

STEP 10

implify the equation.
$g(f(x)) =14x -4 -7$

STEP 11

Further simplify the equation.
$g(f(x)) =14x -11$

STEP 12

Now that we have both $(f \circ g)(x)$ and $(g \circ f)(x)$ , we can compare them to check if they are equivalent.
$f(g(x)) =14x -51$ $g(f(x)) =14x -11$ From the above, we can see that $(f \circ g)(x)$ is not equivalent to $(g \circ f)(x)$ .

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