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Question 10:

Which is not a property of an inverse function? a. The inverse of $f^{-1}(x)$ is $f(x)$.
b. $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f$.
c. $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f^{-1}$.
d. $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$.

Answer: The correct answer is d.
Explanation: Option d states that $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$, which is incorrect. The correct statement should be $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f$.

Question 11:

What is the inverse of $-6x - 5$? a. $-\frac{x+5}{6}$
b. $-\frac{x-5}{6}$
c. $\frac{x+5}{6}$
d. $\frac{x-5}{6}$

Answer: The correct answer is c: $\frac{x+5}{6}$.
Explanation: To find the inverse, switch $x$ and $y$ in the equation $y = -6x - 5$, and solve for $y$: $x = -6y - 5 \implies y = \frac{x + 5}{6}$

Question 12:

What is the inverse of $(x+1)^3 - 1$? a. $\sqrt[3]{x + 1} - 1$
b. $\sqrt[3]{x + 1} + 1$
c. $\sqrt[3]{x + 1 + 1}$
d. $\sqrt[3]{x - 1} + 1$

Answer: The correct answer is d: $\sqrt[3]{x - 1} + 1$.
Explanation: Start by setting $y = (x+1)^3 - 1$, and solve for $x$: [ y + 1 = (x+1)^3 \implies \sqrt[3]{y + 1} = x + 1 \implies x = \sqrt[3]{y + 1} - 1 ]

Question 13:

What is the inverse of $g(x) = 9x + 20$? a. $g^{-1}(x) = \frac{x - 20}{9}$
b. $g^{-1}(x) = \frac{x + 20}{9}$
c. $g^{-1}(x) = \frac{x + 9}{20}$
d. $g^{-1}(x) = \frac{x - 9}{20}$

Answer: The correct answer is a: $g^{-1}(x) = \frac{x - 20}{9}$.
Explanation: To find the inverse, solve the equation $y = 9x + 20$ for $x$: $y - 20 = 9x \implies x = \frac{y - 20}{9}$

Would you like any further details or explanations on these problems?

Relative Questions:

1. How do you generally find the inverse of a function algebraically?
2. What are some properties that hold for all inverse functions?
3. Can every function have an inverse? Why or why not?
4. How can you verify if two functions are inverses of each other?
5. What are some real-world applications of inverse functions?

Tip: When solving for an inverse function, always ensure to swap $x$ and $y$ and then solve for $y$ again to get the inverse expression.