Let's break down the tasks from your image step by step:

Question 8:

Given:

• $f(x) = \frac{x+3}{2x-7}$
• $g(x) = \frac{2x-3}{x+1}$
• $h(x) = \frac{5x+1}{3x-1}$

You are asked to:

1. Evaluate:
   • $f(5)$
   • $g(4)$
2. Show that:
   • $h(f(5)) = \frac{43}{21}$

Solution:

(a) Evaluate $f(5)$ and $g(4)$:

1. For $f(5)$: $f(x) = \frac{x+3}{2x-7}$ Substitute $x = 5$: $f(5) = \frac{5 + 3}{2(5) - 7} = \frac{8}{10 - 7} = \frac{8}{3}$

2. For $g(4)$: $g(x) = \frac{2x - 3}{x + 1}$ Substitute $x = 4$: $g(4) = \frac{2(4) - 3}{4 + 1} = \frac{8 - 3}{5} = \frac{5}{5} = 1$

(b) Show that $h(f(5)) = \frac{43}{21}$:

• First, recall that $f(5) = \frac{8}{3}$.
• Now, substitute $f(5)$ into the function $h(x)$: $h(x) = \frac{5x+1}{3x-1}$ Substitute $x = \frac{8}{3}$ into $h(x)$: $h\left(\frac{8}{3}\right) = \frac{5\left(\frac{8}{3}\right) + 1}{3\left(\frac{8}{3}\right) - 1} = \frac{\frac{40}{3} + 1}{8 - 1} = \frac{\frac{40}{3} + \frac{3}{3}}{7} = \frac{\frac{43}{3}}{7}$ Simplifying further: $h\left(\frac{8}{3}\right) = \frac{43}{21}$ Thus, it is shown that $h(f(5)) = \frac{43}{21}$.

Question 17:

Given:

• $f(x) = 3x - 1$
• $g(x) = 2x$
• $h(x) = x + 1$

You are asked to:

1. Find $h(g(f(x)))$
2. Find $h(g(f(-2)))$

Solution:

(a) Find $h(g(f(x)))$:

1. First, find $f(x)$: $f(x) = 3x - 1$
2. Now, find $g(f(x))$: $g(x) = 2x \quad \text{so} \quad g(f(x)) = 2(3x - 1) = 6x - 2$
3. Finally, find $h(g(f(x)))$: $h(x) = x + 1 \quad \text{so} \quad h(g(f(x))) = (6x - 2) + 1 = 6x - 1$

(b) Find $h(g(f(-2)))$:

1. First, calculate $f(-2)$: $f(-2) = 3(-2) - 1 = -6 - 1 = -7$
2. Now, calculate $g(f(-2))$: $g(-7) = 2(-7) = -14$
3. Finally, calculate $h(g(f(-2)))$: $h(-14) = -14 + 1 = -13$

Would you like to dive deeper into any part of the explanation, or do you have any questions?

Related Questions:

1. How would the result change if $f(x)$, $g(x)$, or $h(x)$ were modified?
2. Can you evaluate $h(g(f(1)))$ using the same method?
3. How would you approach finding $g(h(f(x)))$?
4. What happens to the compositions if the domain of one function is restricted?
5. Can you derive the inverse of $f(x)$ or $g(x)$?

Tip:

When working with composite functions, always start from the innermost function and work your way outward step by step.