1Which of these statements about inverse functions is true? (a) If f: A → B is a function, the inverse function is defined in every case. (b) If f: A → B is a function, the inverse B → A is also a function in every case. (c) If f: A → B is a one-to-one onto function, the inverse A → B is also a function. (d) If f: A → B is a one-to-one onto function, the inverse B → A is also a function.
2If \( f(x) = \{(1, 1), (2, 4), (3, 9), (4, 16)\} \) is a one-to-one onto function, what is \( f^{-1}(x) \)? Choose from: {(1, 1), (4, 2), (3, 9), (16, 4)}; {(1, 1), (4, 2), (9, 3), (16, 4)}; {(1, 1), (4, 2), (3, 9), (4, 16)}; {(1, 1), (2, 4), (3, 9), (16, 4)}.
3If \( f(x) = 4x + 5 \) is a one-to-one onto function, what is \( f^{-1}(x) \)? Choose from: \( 5 - 4x \), \( \frac{x - 4}{5} \), \( \frac{x + 5}{4} \), \( \frac{x - 5}{4} \).
4If \( g(x) = \frac{1 - 3x}{5} \) is a one-to-one onto function, what is \( g^{-1}(x) \)? Choose from: \( \frac{1 + 3x}{5} \), \( \frac{1 - 5x}{5} \), \( \frac{1 - 5x}{3} \), \( \frac{1 + 5x}{3} \).
5If \( h(x) = \frac{1 + 2x}{3} \) is a one-to-one onto function, what is \( h^{-1}(-1) \)? Choose from: -2, -1, 1, 2.
6What is an inverse function? Define it with the help of an arrow diagram.
7If f(x) is a one-to-one onto function and \( f(x) = \{(1, 2), (2, 4), (3, 6)\} \), find \( f^{-1}(x) \) as ordered pairs and show it in an arrow diagram.
8If \( f(x) = 2x - 3 \) is a function, where \( x \in R \) and R is the set of real numbers, is f a one-to-one onto function? If it is, find \( f^{-1}(x) \).
9If \( f(x) = 5x - 2 \) and \( f^{-1}(x) = 2 \), find the value of x.
10If \( g(x) = \frac{3x - 7}{4} \) and \( g^{-1}(x) = 5 \), find the value of x.
11If \( h(x) = \frac{2x - 3}{3x - 5} \) and \( h^{-1}(x) = 2 \), find the value of x.
12If \( f(x) = \frac{x}{3 + 2x} \) and \( f^{-1}(x) = f(x) \), find the value of x.
13If \( f(x) = 7x - 1 \) and \( g(x) = \frac{2x - 3}{5} \) are one-to-one onto functions, find \( g^{-1}f(x) \).
14If \( f(x) = 4x - 3 \) and \( g(x) = \frac{x + 2}{5} \) are one-to-one onto functions, find \( f^{-1}g^{-1}(2) \).
15If \( g(x) = \frac{3x - 1}{x - 1}, x \neq 1 \) and \( h(x) = \frac{2x - 1}{x - 3}, x \neq 3 \), find \( (gh)^{-1}(x) \) and \( (hg)^{-1}(x) \).
16\( f(x) = 2x - 3 \) and \( g(x) = \frac{2x - 7}{3} \) are one-to-one onto functions. If \( ff(x) = g^{-1}(x) \), find the value of x.
17\( f(x) = 2x - 5 \) and \( g(x) = \frac{3x + 5}{2} \) are one-to-one onto functions. If \( ff(x) = g^{-1}(x) \), find the value of x.
18\( f(x) = \frac{x + 1}{2} \) and \( g(x) = \frac{x - 5}{2} \) are one-to-one onto functions. If \( fg^{-1}(x) = 6 \), find the value of x.
19Must a function f(x) be defined as a one-to-one onto function for its inverse \( f^{-1}(x) \) to be defined? Can the inverse \( f^{-1}(x) \) of every function f(x) be defined? Discuss.
20Is \( f^{-1}(x) \) the same as \( \frac{1}{f(x)} \)? Explain with \( f(x) = 2x + 1 \).
21Let \( h(x) = x^2 \) with domain {-2, -1, 1, 2} and codomain {1, 4}. Does h have an inverse function? Give a reason.
22Find the inverse of \( f(x) = \frac{x + 5}{3} \) and check your answer by showing that \( f(f^{-1}(x)) = x \).