The image contains a problem-solving task that asks to find the inverse of several functions. The functions are:

1. $f(x) = 5x + 3$
2. $g(x) = \frac{2x - 5}{3}$
3. $h(x) = 2x^3 - 3$
4. $t(x) = \frac{3x + 1}{2x - 4}$

Let me guide you through the process of finding the inverse for each function:

1. Finding the inverse of $f(x) = 5x + 3$

To find the inverse, follow these steps:

• Replace $f(x)$ with $y$: $y = 5x + 3$
• Solve for $x$: $y - 3 = 5x \quad \Rightarrow \quad x = \frac{y - 3}{5}$
• Replace $y$ with $x$ to get the inverse function: $f^{-1}(x) = \frac{x - 3}{5}$

2. Finding the inverse of $g(x) = \frac{2x - 5}{3}$

• Replace $g(x)$ with $y$: $y = \frac{2x - 5}{3}$
• Solve for $x$: $3y = 2x - 5 \quad \Rightarrow \quad 2x = 3y + 5 \quad \Rightarrow \quad x = \frac{3y + 5}{2}$
• Replace $y$ with $x$: $g^{-1}(x) = \frac{3x + 5}{2}$

3. Finding the inverse of $h(x) = 2x^3 - 3$

• Replace $h(x)$ with $y$: $y = 2x^3 - 3$
• Solve for $x$: $y + 3 = 2x^3 \quad \Rightarrow \quad x^3 = \frac{y + 3}{2} \quad \Rightarrow \quad x = \sqrt[3]{\frac{y + 3}{2}}$
• Replace $y$ with $x$: $h^{-1}(x) = \sqrt[3]{\frac{x + 3}{2}}$

4. Finding the inverse of $t(x) = \frac{3x + 1}{2x - 4}$

• Replace $t(x)$ with $y$: $y = \frac{3x + 1}{2x - 4}$
• Multiply both sides by $2x - 4$: $y(2x - 4) = 3x + 1$ Expand: $2xy - 4y = 3x + 1$
• Get all terms involving $x$ on one side: $2xy - 3x = 4y + 1$ Factor $x$ on the left-hand side: $x(2y - 3) = 4y + 1$
• Solve for $x$: $x = \frac{4y + 1}{2y - 3}$
• Replace $y$ with $x$: $t^{-1}(x) = \frac{4x + 1}{2x - 3}$

Would you like further details or have any questions?

Here are 5 related questions to expand this information:

1. How can you verify if the calculated inverse function is correct?
2. What types of functions do not have an inverse, and why?
3. How does the graphical interpretation of a function's inverse look on the Cartesian plane?
4. Are there any restrictions on the domains or ranges of the inverse functions obtained?
5. Can all polynomial functions have inverses? Why or why not?

Tip:

Always check if the original function is one-to-one before finding its inverse, as only one-to-one functions have inverses.