Let's solve the given problems step by step.

Given Functions:

$f(x) = 3x$ $g(x) = 5 - 2x$

1. Find $(f \circ g)(x)$

$(f \circ g)(x) = f(g(x))$ Substituting $g(x)$: $f(g(x)) = f(5 - 2x) = 3(5 - 2x) = 15 - 6x$ Thus, $(f \circ g)(x) = 15 - 6x$

2. Find $(f \circ g)(5)$

Using the result from the previous part: $(f \circ g)(5) = 15 - 6(5) = 15 - 30 = -15$

3. Find $(g \circ f)(x)$

$(g \circ f)(x) = g(f(x))$ Substituting $f(x)$: $g(f(x)) = g(3x) = 5 - 2(3x) = 5 - 6x$ Thus, $(g \circ f)(x) = 5 - 6x$

4. Find $(g \circ f)(10)$

Using the result from the previous part: $(g \circ f)(10) = 5 - 6(10) = 5 - 60 = -55$

5. Find $(f \circ f)(x)$

$(f \circ f)(x) = f(f(x))$ Substituting $f(x)$: $f(f(x)) = f(3x) = 3(3x) = 9x$ Thus, $(f \circ f)(x) = 9x$

6. Find $(f \circ f)(20)$

Using the result from the previous part: $(f \circ f)(20) = 9(20) = 180$

7. Find $(g \circ g)(x)$

$(g \circ g)(x) = g(g(x))$ Substituting $g(x)$: $g(g(x)) = g(5 - 2x) = 5 - 2(5 - 2x) = 5 - 10 + 4x = 4x - 5$ Thus, $(g \circ g)(x) = 4x - 5$

8. Find $(g \circ g)(3)$

Using the result from the previous part: $(g \circ g)(3) = 4(3) - 5 = 12 - 5 = 7$

Summary of Results

1. $(f \circ g)(x) = 15 - 6x$
2. $(f \circ g)(5) = -15$
3. $(g \circ f)(x) = 5 - 6x$
4. $(g \circ f)(10) = -55$
5. $(f \circ f)(x) = 9x$
6. $(f \circ f)(20) = 180$
7. $(g \circ g)(x) = 4x - 5$
8. $(g \circ g)(3) = 7$

Would you like more details or have any questions?

5 Related Questions

1. What is the definition of function composition in mathematics?
2. How do you interpret the meaning of $(f \circ g)(x)$ in real-world applications?
3. How can you determine whether the composition of two functions is commutative, i.e., whether $f \circ g = g \circ f$?
4. What happens when you compose more than two functions together, e.g., $(f \circ g \circ h)(x)$?
5. How do inverses of functions relate to function composition?

Tip

When composing functions, always substitute the inner function into the outer function carefully, and work step-by-step to avoid mistakes.