Math  /  Algebra

QuestionFind (fg)(x)(f \circ g)(x) f(x)=6xg(x)=x+9(fg)(x)=\begin{aligned} f(x) & =6 x \\ g(x) & =x+9 \\ (f \circ g)(x)= & \end{aligned}

Studdy Solution

STEP 1

What is this asking? We need to find the *composition* of two functions, f(x)f(x) and g(x)g(x), which basically means plugging g(x)g(x) into f(x)f(x). Watch out! Don't plug f(x)f(x) into g(x)g(x)!
The order matters in function composition!

STEP 2

1. Substitute g(x)g(x) into f(x)f(x)
2. Simplify the expression

STEP 3

We're given f(x)=6xf(x) = 6x and g(x)=x+9g(x) = x + 9.
We want to find (fg)(x)(f \circ g)(x), which means we take g(x)g(x) and put it *inside* f(x)f(x) wherever we see an xx.
Think of f(x)f(x) as a tiny machine that takes an input (xx), multiplies it by **6**, and spits out the result.
Now, instead of just xx going in, we're feeding the entire function g(x)g(x) into the machine!

STEP 4

So, we replace the xx in f(x)f(x) with the *entire* expression for g(x)g(x). (fg)(x)=f(g(x))=f(x+9) (f \circ g)(x) = f(g(x)) = f(x+9) Now, remember, f(x)=6xf(x) = 6 \cdot x, so f(something)=6(something)f(\text{something}) = 6 \cdot (\text{something}).
In our case, "something" is x+9x+9, so: (fg)(x)=6(x+9) (f \circ g)(x) = 6 \cdot (x+9)

STEP 5

We **distribute** the **6** across the terms inside the parentheses: (fg)(x)=6x+69 (f \circ g)(x) = 6 \cdot x + 6 \cdot 9

STEP 6

(fg)(x)=6x+54 (f \circ g)(x) = 6x + 54 And there we have it!
A beautifully simplified expression!

STEP 7

(fg)(x)=6x+54(f \circ g)(x) = 6x + 54

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