Question

Consider the functions f(x) = ln(x) - 4; g(x) = ex+7; h(x) = 9(2x); j(x) = 6x - 1

a) (f•g)(x)
b) (j•h)(x)

Answer 1

The composite function (f•g)(x) equals x + 3, using the inverse relationship between ln and e. The composite function (j•h)(x) equals 54·
`2^x`- 1, which is derived by substituting h(x) into j(x).

The student has asked to find the composite functions (f•g)(x) and (j•h)(x) given the functions f(x) = ln(x) - 4; g(x) =
`e^(x+7)`; h(x) = 9(
`2^x`); j(x) = 6x - 1.

Composite Function (f•g)(x)

To find (f•g)(x), we substitute g(x) into f(x), yielding f(g(x)) = f(
`e^(x+7)`) = ln(
`e^(x+7)`) - 4. Using the property that the natural logarithm and exponential functions are inverses, ln(
`e^y`) = y, we simplify this to f(g(x)) = (x+7) - 4 = x + 3.

Composite Function (j•h)(x)

To find (j•h)(x), we substitute h(x) into j(x), giving us j(h(x)) = j(9·
`2^x`) = 6(9·
`2^x`) - 1. This expression simplifies to (j•h)(x) = 54·
`2^x`- 1.