Math Snap

STEP 1

What is this asking?
We're given two functions, $f(x)$ and $g(x)$, and we need to find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$, as well as their values at $x = 6$.
Watch out!
Remember the order matters in function composition! $(f \circ g)(x)$ means $f(g(x))$, not $g(f(x))$.

STEP 2

1. Find $(f \circ g)(x)$
2. Find $(g \circ f)(x)$
3. Find $(f \circ g)(6)$
4. Find $(g \circ f)(6)$

STEP 3

Let's start with $(f \circ g)(x)$, which means we're putting $g(x)$ inside $f(x)$.
It's like a function turducken!

STEP 4

We know $f(x) = 8x - 7$ and $g(x) = \frac{x + 7}{8}$.
So, $(f \circ g)(x) = f(g(x))$.
This means wherever we see an $x$ in $f(x)$, we replace it with the entire $g(x)$ function.

STEP 5

$$f(g(x)) = f\left(\frac{x + 7}{8}\right) = 8 \cdot \left(\frac{x + 7}{8}\right) - 7$$

STEP 6

Now, we simplify:
$$8 \cdot \left(\frac{x + 7}{8}\right) - 7 = \frac{8}{8} \cdot (x+7) - 7 = 1 \cdot (x + 7) - 7 = x + 7 - 7 = x$$ So, $(f \circ g)(x) = x$!

STEP 7

Now, let's find $(g \circ f)(x)$, which means $g(f(x))$.
This time, $f(x)$ goes inside $g(x)$.

STEP 8

We have $g(x) = \frac{x + 7}{8}$ and $f(x) = 8x - 7$.
So,
$$g(f(x)) = g(8x - 7) = \frac{(8x - 7) + 7}{8}$$

STEP 9

Let's simplify this:
$$\frac{8x - 7 + 7}{8} = \frac{8x}{8} = \frac{8}{8} \cdot x = 1 \cdot x = x$$ So, $(g \circ f)(x) = x$!

STEP 10

We already found that $(f \circ g)(x) = x$.
So, to find $(f \circ g)(6)$, we just substitute $x = 6$:
$$(f \circ g)(6) = 6$$

STEP 11

Similarly, we found $(g \circ f)(x) = x$.
So, for $(g \circ f)(6)$, we substitute $x = 6$:
$$(g \circ f)(6) = 6$$

a. $(f \circ g)(x) = x$
b. $(g \circ f)(x) = x$
c. $(f \circ g)(6) = 6$
d. $(g \circ f)(6) = 6$