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Problem 1:

You are given a set of ordered pairs $(8,3), (5,6), (-2,-3), (0,1), (7,12)$. The task is to find the inverse of the function described by these pairs.

Solution:

The inverse of a function swaps the x-values with the y-values. Therefore, the inverse set of ordered pairs would be: $(3,8), (6,5), (-3,-2), (1,0), (12,7)$

Problem 2:

You need to determine if $f^{-1}(x) = \frac{x - 2}{5}$ is the inverse of $f(x) = 5x + 2$.

Solution:

To check this, let's follow two steps:

1. Find the inverse of $f(x) = 5x + 2$.
2. Compare the result with $f^{-1}(x) = \frac{x - 2}{5}$.

To find the inverse, set $y = 5x + 2$ and solve for $x$: $y = 5x + 2 \implies x = \frac{y - 2}{5}$ Thus, $f^{-1}(x) = \frac{x - 2}{5}$, which matches the given function.

So, yes, $f^{-1}(x) = \frac{x - 2}{5}$ is the inverse of $f(x) = 5x + 2$.

Problem 3:

You are asked to find the inverse of $f(x) = \frac{3x - 2}{2x + 10}$.

Solution:

To find the inverse:

1. Set $y = \frac{3x - 2}{2x + 10}$ and solve for $x$.
2. Multiply both sides by $(2x + 10)$ to get rid of the denominator: $y(2x + 10) = 3x - 2$
3. Expand: $2xy + 10y = 3x - 2$
4. Collect all terms with $x$ on one side: $2xy - 3x = -2 - 10y$
5. Factor out $x$: $x(2y - 3) = -2 - 10y$
6. Solve for $x$: $x = \frac{-2 - 10y}{2y - 3}$ Thus, the inverse of $f(x) = \frac{3x - 2}{2x + 10}$ is: $f^{-1}(x) = \frac{-2 - 10x}{2x - 3}$

Would you like more details on any of these solutions?

5 Related Questions:

1. How do you find the inverse of any general function?
2. What are the key differences between injective, surjective, and bijective functions?
3. How can you verify if two functions are inverses of each other?
4. Can all functions have an inverse? Why or why not?
5. How do domain and range change when finding the inverse of a function?

Tip:

When solving for the inverse of a function, always ensure that the original function is one-to-one, meaning that each input has a unique output. Otherwise, the inverse won't be valid.