QuestionFind $f(x)$ if $(f \circ g)(x) = 3x + 5$ and $g(x) = 2x - 1$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(g(x))$ is given as $3x+5$ . The function $g(x)$ is given as $x-1$
3. We are asked to find the function $f(x)$

STEP 2

First, we need to understand what $f(g(x))$ means. In function composition, $f(g(x))$ means that we first apply the function $g(x)$ to $x$ , and then apply the function $f(x)$ to the result.

STEP 3

To find $f(x)$ , we need to express $f(g(x))$ in terms of $g(x)$ . We can do this by substituting $g(x)$ into $f(g(x))$ .
$f(g(x)) = f(2x-1) =3x+5$

STEP 4

Now, we can see that $f(g(x))$ is equal to $f(2x-1)$ , which is equal to $3x+$ . So, we can say that $f(2x-1) =3x+$ .

STEP 5

To find $f(x)$ , we need to express $f(2x-1)$ in terms of $x$ . We can do this by replacing $2x-1$ with $x$ in the equation $f(2x-1) =3x+5$ .
$f(x) =3\left(\frac{x+1}{2}\right) +5$

STEP 6

Now, simplify the equation to find $f(x)$ .
$f(x) = \frac{3x+3}{2} +5$

STEP 7

Further simplify the equation to find $f(x)$ .
$f(x) = \frac{3x}{2} + \frac{3}{2} +5$

STEP 8

Combine like terms to find $f(x)$ .
$f(x) = \frac{3x}{2} + \frac{13}{2}$ So, the function $f(x)$ is $f(x) = \frac{3x}{2} + \frac{13}{2}$ .

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