1Which one of these statements about composite functions is wrong? (a) If f: A to B and g: B to C are functions and the relation A to C is also a function, it is a composite function. (b) If g: A to B and f: B to C are one to one and onto functions, the function A to C is a composite function. (c) If g: A to B and f: B to C are defined, the function A to C is written gof. (d) If f: A to B and g: B to C are defined, the relation C to A is not the composite of f and g.
2In an arrow diagram, g carries set A into set B and f carries set B into set C. Which of these denotes the function from A to C: fg(x), gf(x), ff(x) or gg(x)?
3If f(x) = {(3, -1), (2, 0), (1, 1)} and g(x) = {(-1, 4), (0, 5), (1, 6)}, what is gf(x)? Choose from {(4, 3), (2, 5), (1, 6)}, {(3, 4), (2, 5), (2, 6)}, {(3, 4), (1, 5), (2, 6)} and {(3, 4), (2, 5), (1, 6)}.
4If f(x) = 3x - 1 and g(x) = x + 1, what is fg(x)? Choose from 3x - 1, 3x + 1, 3x - 2 and 3x + 2.
5If f(x) = 1 - 3x and g(x) = 2x + 1, what is gf(1/2)? Choose from 3, 1, 2 and 0.
6Define a composite function with the help of an arrow diagram.
7In an arrow diagram, g: A to B and f: B to C, with A = {1, 2, 3, 4}, B = {3, 4, 2, 5} and C = {0, 1, -1, 2}. The arrows of g send 1 to 2, 2 to 3, 3 to 4 and 4 to 5. The arrows of f send 3 to -1, 4 to 0, 2 to -1 and 5 to 1. (a) Write g as ordered pairs. (b) Write f as ordered pairs. (c) Find fog(1), fog(2), fog(3) and fog(4). (d) Write the composite fog as ordered pairs.
8If f(x) = {(1, 2), (2, 3), (3, 4)} and g(x) = {(2, 4), (3, 5), (4, 6)}, show gf(x) in an arrow diagram and write it as ordered pairs.
9If f(x) = 3x + 2 and g(x) = 5 - 3x, find (a) gf(x) (b) fg(x) (c) ff(x) (d) ff(3).
10If \( f(x) = \frac{2x - 1}{x + 3} \), where \( x \neq -3 \), and \( g(x) = \frac{3x + 2}{x - 4} \), where \( x \neq 4 \), find fg(x).
11If f(x) = 2x + 5 and \( g(x) = \frac{x - 5}{2} \), prove that fg(x) is an identity function.
12If \( f(x) = \frac{x - 3}{2} \) and fg(x) = x, find g(x).
13If g(x) = 4 - x and gh(x) = 11 - 2x, find h(x).
14If \( f(x) = \frac{6}{x - 2} \), where \( x \neq 2 \), \( g(x) = ax^2 - 1 \) and gf(5) = 7, find the value of a.
15If f(x) = ax + 5, g(x) = 8x + 13 and gf(5) = 93, find the value of a.
16If f(x) = kx + 1, g(x) = 3x - 7 and gf(1) = 2, find the value of k.
17If f(x) = 3x - 2, fg(x) = 6x - 2 and gf(x) = 8, find the value of x.
18If the functions g: A to B and f: B to C are not properly defined onto functions, why is the composite fg: A to C not defined? Discuss.
19If f(x) = x + 2 and g(x) = x², find fg(x) and gf(x). Are they equal?
20If g(x) = 2x + 1, find gg(x) and gg(2).
21If f(x) = 3x and g(x) = 2x + 1, find gf(2) in two different ways.