QuestionFind the value of $(f \circ g)(2)$ $f(x) = 5x + 2$ and $g(x) = 3x - 4$

Studdy Solution

STEP 1

What is this asking? We need to find the value of a composite function at a specific point, meaning we'll plug a number into one function, then plug that result into another function! Watch out! Remember the order matters with composite functions! $(f \circ g)(x)$ means we apply $g$ first, then $f$ .
It's like a tiny assembly line for numbers!

STEP 2

1. Evaluate $g(2)$
2. Evaluate $f(g(2))$

STEP 3

Let's kick things off by evaluating $g(2)$ .
We're doing this because $g$ is the inner function in our composite function.
Remember, we work from the inside out!

STEP 4

We know that $g(x) = 3x - 4$ , so we substitute $x$ with our input value, which is $2$ : $g(2) = 3 \cdot 2 - 4$

STEP 5

Now, we simplify: $g(2) = 6 - 4$ $g(2) = 2$ So, $g(2)$ equals 2!
This intermediate result is what we'll plug into $f$ next.

STEP 6

We just found that $g(2) = 2$ , so $(f \circ g)(2)$ is the same as $f(g(2))$ , which is the same as $f(2)$ .
We're doing this because we've simplified the inside of the composite function, and now we're ready to tackle the outside function, $f$ .

STEP 7

We know that $f(x) = 5x + 2$ , so let's substitute $x$ with our new input, which is $2$ : $f(2) = 5 \cdot 2 + 2$

STEP 8

Time to simplify: $f(2) = 10 + 2$ $f(2) = 12$ So, $f(2)$ , which is the same as $f(g(2))$ , equals 12!

STEP 9

$(f \circ g)(2) = 12$

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