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

Studdy Solution

STEP 1

Assumptions1. The function composition $(f \circ g)(x)$ is given as $3x +5$ . The function $g(x)$ is given as $x -1$
3. We need to find the function $f(x)$

STEP 2

We know that $(f \circ g)(x) = f[g(x)]$ . This means that we apply the function $g(x)$ first, and then apply the function $f(x)$ to the result.

STEP 3

Given $g(x) =2x -1$ , we substitute this into the function composition to get $f[g(x)] = f[2x -1]$

STEP 4

We know that $f[2x -1] =3x +$ . This means that wherever we see $2x -1$ in the function $f$ , we should be able to replace it with $3x +$ to get the same result.

STEP 5

Let's assume that $f(x) = ax + b$ . We can find the values of $a$ and $b$ by comparing $f[2x -1]$ with $3x +5$ .

STEP 6

Substitute $x$ with $2x -1$ in $f(x) = ax + b$ to get $f[2x -1] = a(2x -1) + b$ .

STEP 7

implify $f[2x -1] = a(2x -1) + b$ to get $f[2x -1] =2ax - a + b$ .

STEP 8

We know that $f[2x -1] =3x +5$ and $f[2x -1] =2ax - a + b$ . So, we can set these two equal to each other to find the values of $a$ and $b$ .

STEP 9

Set $2ax - a + b =3x +5$ .

STEP 10

We can compare the coefficients on the left and right side of the equation to find the values of $a$ and $b$ .

STEP 11

Comparing the coefficients of $x$ , we get $a =3$ . So, $a =3/$ .

STEP 12

Comparing the constants, we get $-a + b =5$ . Substitute $a =/2$ into this equation to get $-/2 + b =5$ .

STEP 13

olve for $b$ to get $b =5 +3/2 =13/2$ .

STEP 14

Now that we have the values of $a$ and $b$ , we can write the function $f(x)$ as $f(x) =3/2x +13/2$ .
So, the function $f(x)$ that satisfies the given conditions is $f(x) =3/2x +13/2$ .

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