2.3 b. h⁻¹(3) = -27
2.4 a. h(g(2)) = 6
b. f(h(x) = 5·x² + 10
3. a. f(g(2) = -1.5
b. g(f(5)) = 2
The steps that can be used to find the inverse of the function and the values of the composite functions are presented as follows;
b. -0.2·x - 2.4 = y
The inverse of the above function can be found by making x the subject of the equation as follows;
-0.2·x = y + 2.4
x = (1/-0.2)·y + (2.4/(-0.2))
x = -5·y - 12
The inverse of a function is the value of the input of the function that produces a specified output, therefore, the inverse of a function, takes the output, y, and produces the corresponding input value, x
h⁻¹(x) = -5·x - 12
h⁻¹(3) = -5 × 3 - 12
-5 × 3 - 12 = -27
h⁻¹(3) = -27
2. f(x) = 5·x, h(x) = x² + 2, g(x) = x - 4
h(g(2))
g(2) = 2 - 4
g(2) = -2
h(g(2)) = h(-2)
h(-2) = (-2)² + 2
(-2)² + 2 = 6
h(-2) = 6
h(g(2)) = 6
b. f(h(x))
h(x) = x² + 2
f(h(x) = 5 × (x² + 2)
5 × (x² + 2) = 5·x² + 10
f(h(x)) = 5·x² + 10
3. a. The table of values indicates that we get; g(2) = 7
f(g(2)) = f(7)
The values of the ordered pairs that can be obtained from the graph indicates that we get; (7, -1.5)
Therefore; f(7) = -1.5
f(g(2)) = -1.5
b. g(f(5))
The values of the ordered pairs that can be obtained from the graph indicates that we get; (5, 0)
Therefore; f(5) = 0, which indicates, g(f(5)) = g(0)
The ordered pair of values in the table, (0, 2) indicates that we get;
g(0) = 2, therefore; g(f(5)) = 2