Let's go through each question step-by-step.

Question 1

1. (a) Find $f^{-1}(x)$: We need to find the inverse of $f(x)$.

   Given: $f(x) = 3x - 1$

   1. Set $f(x) = y$: $y = 3x - 1$
   2. Solve for $x$: $y + 1 = 3x$ $x = \frac{y + 1}{3}$
   3. Replace $y$ with $x$ in the expression: $f^{-1}(x) = \frac{x + 1}{3}$

2. (b) Calculate $f^{-1}(x)$: We already found $f^{-1}(x) = \frac{x + 1}{3}$.

Question 2

1. (a) Calculate the value of $f(g(2))$: Given $f(x) = 3x - 1$ and $g(x) = 2x + 4$.

   1. First, find $g(2)$: $g(2) = 2(2) + 4 = 4 + 4 = 8$
   2. Now, find $f(g(2)) = f(8)$: $f(8) = 3(8) - 1 = 24 - 1 = 23$

   So, $f(g(2)) = 23$.

2. (b) Calculate the value of $f(f(3))$:

   1. First, find $f(3)$: $f(3) = 3(3) - 1 = 9 - 1 = 8$
   2. Now, find $f(f(3)) = f(8)$ (we already calculated this in part (a)): $f(8) = 23$

   So, $f(f(3)) = 23$.

3. (c) Find $gf(x)$:

   Here, $gf(x) = g(f(x))$.

   1. Start with $f(x) = 3x - 1$.
   2. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(3x - 1) = 2(3x - 1) + 4$
   3. Simplify: $= 6x - 2 + 4 = 6x + 2$

   So, $gf(x) = 6x + 2$.

Question 3

1. (a) Find $f(g(x))$: Given $f(x) = x^2 - 3$, $g(x) = 2x + 1$, and $h(x) = \frac{x}{2}$.

   1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(2x + 1) = (2x + 1)^2 - 3$
   2. Expand: $= 4x^2 + 4x + 1 - 3 = 4x^2 + 4x - 2$

   So, $f(g(x)) = 4x^2 + 4x - 2$.

2. (b) Find $g(h(x))$:

   1. Substitute $h(x)$ into $g(x)$: $g(h(x)) = g\left(\frac{x}{2}\right) = 2\left(\frac{x}{2}\right) + 1$
   2. Simplify: $= x + 1$

   So, $g(h(x)) = x + 1$.

3. (c) Find $h^{-1}(x)$:

   1. Start with $h(x) = \frac{x}{2}$.
   2. Set $h(x) = y$: $y = \frac{x}{2}$.
   3. Solve for $x$: $x = 2y$
   4. Replace $y$ with $x$: $h^{-1}(x) = 2x$

Question 4

1. (a) Solve $f(x) = 17$:

   Given $f(x) = 4x - 7$.

   1. Set $f(x) = 17$: $4x - 7 = 17$
   2. Add 7 to both sides: $4x = 24$
   3. Divide by 4: $x = 6$

   So, $x = 6$ when $f(x) = 17$.

Would you like further explanation on any specific part? Here are some related questions you might consider exploring:

1. How can we verify if two functions are inverses of each other?
2. What is the difference between composite functions like $f(g(x))$ and $g(f(x))$?
3. How do we find the inverse of a quadratic function?
4. How can we solve equations with more than one function involved?
5. What are some real-life applications of composite and inverse functions?

Tip: When calculating $f(g(x))$ or $g(f(x))$, always simplify your result as much as possible to avoid complex expressions.