Let's work through the problems step-by-step:

Problem 3:

Given the functions: $F(x) = 4x - 2$ $g(x) = x + 5$

a. (F ∘ g)(x): This means $F(g(x))$, or applying $g(x)$ inside $F(x)$.

$(F ∘ g)(x) = F(g(x)) = F(x + 5) = 4(x + 5) - 2$ $= 4x + 20 - 2 = 4x + 18$ So, $(F ∘ g)(x) = 4x + 18$.

b. (F ∘ g)(-1): Substitute $x = -1$ into $(F ∘ g)(x)$.

$(F ∘ g)(-1) = 4(-1) + 18 = -4 + 18 = 14$

So, $(F ∘ g)(-1) = 14$.

c. (g ∘ F)(x): This means $g(F(x))$, or applying $F(x)$ inside $g(x)$.

$(g ∘ F)(x) = g(F(x)) = g(4x - 2) = (4x - 2) + 5 = 4x + 3$

So, $(g ∘ F)(x) = 4x + 3$.

d. (g ∘ F)(2): Substitute $x = 2$ into $(g ∘ F)(x)$.

$(g ∘ F)(2) = 4(2) + 3 = 8 + 3 = 11$

So, $(g ∘ F)(2) = 11$.

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Problem 4:

Given the functions: $f(x) = x^2 + 3x - 5$ $g(x) = x + 2$

A. f(2): Substitute $x = 2$ into $f(x)$.

$f(2) = (2)^2 + 3(2) - 5 = 4 + 6 - 5 = 5$

So, $f(2) = 5$.

B. (g ∘ f)(x): This means $g(f(x))$.

$(g ∘ f)(x) = g(f(x)) = f(x) + 2 = (x^2 + 3x - 5) + 2 = x^2 + 3x - 3$

So, $(g ∘ f)(x) = x^2 + 3x - 3$.

C. (g ∘ f)(2): Substitute $x = 2$ into $(g ∘ f)(x)$.

$(g ∘ f)(2) = (2)^2 + 3(2) - 3 = 4 + 6 - 3 = 7$

So, $(g ∘ f)(2) = 7$.

D. (F ∘ g)(x): This means $F(g(x))$, but since $F(x)$ is not given, we can assume it refers to applying $f(g(x))$.

$(f ∘ g)(x) = f(g(x)) = f(x + 2) = (x + 2)^2 + 3(x + 2) - 5$ $= (x^2 + 4x + 4) + 3x + 6 - 5 = x^2 + 7x + 5$

So, $(f ∘ g)(x) = x^2 + 7x + 5$.

E. (f ∘ g)(1): Substitute $x = 1$ into $(f ∘ g)(x)$.

$(f ∘ g)(1) = (1)^2 + 7(1) + 5 = 1 + 7 + 5 = 13$

So, $(f ∘ g)(1) = 13$.

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Would you like further clarification or more details on any part of the solution?

Follow-up questions:

1. How do you find the composition of two functions in general?
2. What is the difference between $(f ∘ g)(x)$ and $(g ∘ f)(x)$?
3. How does substitution work when composing functions?
4. Why is it important to simplify expressions after substituting values into functions?
5. How do we interpret the composition of functions in real-world contexts?

Tip:

Always check the domain of the functions before composing them to ensure that the operations are valid for all values of $x$.