https://file.aiquickdraw.com/mathbot/file/17286743942985lnwnybo.jpg

Let's solve the composition of functions problem displayed in the image. The given functions are:

- \( f(x) = 7x + 10 \)
- \( g(x) = -8x - 3 \)

### 1. \( (f \circ f)(x) = f(f(x)) \)
We first find \( f(f(x)) \), which means substituting \( f(x) \) into \( f \).

- \( f(x) = 7x + 10 \), so substitute \( f(x) \) into \( f \):
- \( f(f(x)) = f(7x + 10) = 7(7x + 10) + 10 = 49x + 70 + 10 = 49x + 80 \)

Thus, \( (f \circ f)(x) = 49x + 80 \).

### 2. \( (f \circ g)(x) = f(g(x)) \)
We now find \( f(g(x)) \), meaning substituting \( g(x) \) into \( f \).

- \( g(x) = -8x - 3 \), so substitute \( g(x) \) into \( f \):
- \( f(g(x)) = f(-8x - 3) = 7(-8x - 3) + 10 = -56x - 21 + 10 = -56x - 11 \)

Thus, \( (f \circ g)(x) = -56x - 11 \).

### 3. \( (g \circ f)(x) = g(f(x)) \)
Next, find \( g(f(x)) \), meaning substituting \( f(x) \) into \( g \).

- \( f(x) = 7x + 10 \), so substitute \( f(x) \) into \( g \):
- \( g(f(x)) = g(7x + 10) = -8(7x + 10) - 3 = -56x - 80 - 3 = -56x - 83 \)

Thus, \( (g \circ f)(x) = -56x - 83 \).

### 4. \( (g \circ g)(x) = g(g(x)) \)
Finally, find \( g(g(x)) \), meaning substituting \( g(x) \) into \( g \).

- \( g(x) = -8x - 3 \), so substitute \( g(x) \) into \( g \):
- \( g(g(x)) = g(-8x - 3) = -8(-8x - 3) - 3 = 64x + 24 - 3 = 64x + 21 \)

Thus, \( (g \circ g)(x) = 64x + 21 \).

### Final Answers:
1. \( (f \circ f)(x) = 49x + 80 \)
2. \( (f \circ g)(x) = -56x - 11 \)
3. \( (g \circ f)(x) = -56x - 83 \)
4. \( (g \circ g)(x) = 64x + 21 \)

Would you like further explanation or have any other questions?

Here are 5 related questions for practice:
1. What is \( (f \circ g)(2) \) for the given functions \( f(x) \) and \( g(x) \)?
2. Find \( (g \circ g)(-1) \) for the given function \( g(x) \).
3. If \( h(x) = 2x^2 + 3x \), what is \( (f \circ h)(x) \)?
4. What is the inverse of the function \( f(x) = 7x + 10 \)?
5. Can you find \( (g \circ f)(0) \) using the given functions?

**Tip:** When composing functions, always substitute the entire expression of the inner function into the outer function.

Consider the functions f(x) = 7x + 10 and g(x) = -8x - 3. Determine the following compositions of functions: (f o f)(x), (f o g)(x), (g o f)(x), and (g o g)(x).

Learn how to compute the composition of functions for f(x) = 7x + 10 and g(x) = -8x - 3. This problem involves finding f o f, f o g, g o f, and g o g. Step-by-step solutions provided.

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