Let's address your questions one by one:

1. Which of the following functions have an inverse?

For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). For injectivity, the function must pass the horizontal line test (no horizontal line crosses the graph more than once). Let's examine each function:

• (a) $f(x) = 2x^3 - 5$:
  This is a cubic function, which is one-to-one because it passes the horizontal line test. Hence, it has an inverse.

• (b) $g(x) = 3x - 8$:
  This is a linear function, and linear functions are always one-to-one (bijective), so it has an inverse.

• (c) $h(x) = \frac{1}{x^2}$:
  This is not one-to-one, as for both $x$ and $-x$, the output will be the same (i.e., the graph is symmetric). Hence, it does not have an inverse.

• (d) $k(x) = |x|$:
  The absolute value function is also not one-to-one because both positive and negative values of $x$ give the same output. Hence, it does not have an inverse.

• (0) $t(x) = x^2 - 6x$:
  This is a quadratic function, and quadratic functions are not one-to-one because they fail the horizontal line test. Hence, it does not have an inverse.

Conclusion: Functions (a) and (b) have an inverse.

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2. Find $f(x) = -x^3 + 2$

Here, you are simply asked to write the function, and it is given as: $f(x) = -x^3 + 2$

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3. Find $f(x)$ if $f^{-1}(x) = \frac{1}{x - 2}$

To find $f(x)$ from $f^{-1}(x)$, we will use the fact that for any $f(x)$ and its inverse $f^{-1}(x)$, the following relation holds: $f(f^{-1}(x)) = x$ Given $f^{-1}(x) = \frac{1}{x - 2}$, we can find $f(x)$ by solving for $f^{-1}(x)$.

Let $y = f^{-1}(x) = \frac{1}{x - 2}$.
To find $f(x)$, solve for $x$ in terms of $y$: $y = \frac{1}{x - 2}$ $y(x - 2) = 1$ $x - 2 = \frac{1}{y}$ $x = \frac{1}{y} + 2$ So, $f(x)$ is the inverse of this expression, which is: $f(x) = \frac{1}{x} + 2$

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Do you have any questions or want more details?

Related Questions:

1. How do we check if a function is injective or surjective?
2. Can a function have an inverse if it's only injective but not surjective?
3. What is the horizontal line test and how is it used to find inverses?
4. How can we find the inverse of a composite function?
5. How do transformations of basic functions affect their inverses?

Tip:

When dealing with inverse functions, always remember that the function and its inverse "undo" each other: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.