Let's work through the problem shown in the image.

We are given:

• $f(x) = \frac{1}{x - 2}$
• $g(x) = 4x + 7$

We need to find:

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

Solution for $(f \circ g)(x)$

The composition $(f \circ g)(x)$ means $f(g(x))$. We first substitute $g(x)$ into $f(x)$:

$(f \circ g)(x) = f(g(x)) = f(4x + 7)$

Substituting $4x + 7$ into $f(x) = \frac{1}{x - 2}$:

$f(4x + 7) = \frac{1}{(4x + 7) - 2} = \frac{1}{4x + 5}$

So, $(f \circ g)(x) = \frac{1}{4x + 5}$.

Solution for $(f \circ g)(3)$

Now, let's calculate $(f \circ g)(3)$:

1. First, find $g(3)$:

$g(3) = 4(3) + 7 = 12 + 7 = 19$

2. Next, substitute 19 into $f(x)$:

$f(19) = \frac{1}{19 - 2} = \frac{1}{17}$

So, $(f \circ g)(3) = \frac{1}{17}$.

Summary

• $(f \circ g)(3) = \frac{1}{17}$
• $(f \circ g)(x) = \frac{1}{4x + 5}$

Would you like any further explanation or details on how the function composition works?

Here are 5 related questions for further exploration:

1. What is the domain of $(f \circ g)(x)$?
2. How would you find $(g \circ f)(x)$ using the given functions?
3. What happens to $(f \circ g)(x)$ if $f(x)$ or $g(x)$ is modified?
4. How can you determine whether $(f \circ g)(x)$ has any vertical asymptotes?
5. What is the inverse of $(f \circ g)(x)$, if it exists?

Tip: Remember that the domain of a composition function is determined by the domain of $g(x)$ and any additional restrictions imposed by $f(x)$ after substituting $g(x)$.