Question

Suppose that the functions g and h are defined for all real numbers x as follows. g (x) = x – 1 h (x) = 3x + 5 Write the expressions for (g • h) (x) and (g + h) (x) and evaluate (g – h) (3).

Answer 1

(g - h)(3) = -11.

Explanation:

To find the expressions for (g • h) (x) and (g + h) (x), we need to use the definitions of function composition and function addition:

(g • h) (x) = g(h(x))

This means we first evaluate h(x), and then use the result as the input for g.

(g + h) (x) = g(x) + h(x)

This means we evaluate g(x) and h(x) separately, and then add their results together.

Using the definitions of g(x) and h(x) given in the problem, we can find:

(g • h) (x) = g(h(x)) = g(3x + 5) = (3x + 5) - 1 = 3x + 4

(g + h) (x) = g(x) + h(x) = (x - 1) + (3x + 5) = 4x + 4

To evaluate (g - h)(3), we need to use the definition of function subtraction:

(g - h)(x) = g(x) - h(x)

This means we evaluate g(x) and h(x) separately, and then subtract their results.

Using the definitions of g(x) and h(x) given in the problem, we can find:

(g - h)(3) = g(3) - h(3) = (3 - 1) - (3(3) + 5) = -11

Therefore, (g - h)(3) = -11.