Need help simplifying the difference quotient!?

Question

Show work please!

(f(x+h)-f(x)) / h
f(x) = 5x - 3
f(x) = -6x + 8
f(x) = 5
f(x) x^2-7x+5

Answer 1

The difference quotient = [f(x + h) - f(x)]/h

• The function f(x) = 5x - 3

f(x + h) = 5(x + h) - 3 = 5x + 5h - 3

f(x + h) - f(x) = (5x + 5h - 3) - (5x - 3)

                    = 5x + 5h - 3 - 5x + 3

                    = 5h

The difference quotient = (5h)/h = 5

The difference quotient = 5.

• The function f(x) = - 6x + 8

f(x + h) = - 6(x + h) + 8 = - 6x - 6h + 8

f(x + h) - f(x) = (- 6x - 6h + 8) - (- 6x + 8)

                    = - 6x - 6h + 8 + 6x - 8

                    = - 6h

The difference quotient = (- 6h)/h = - 6

The difference quotient = - 6.

• The function f(x) = 5

f(x + h) = 5

f(x + h) - f(x) = (5) - (5)

                    = 0

The difference quotient = (0)/h = 0

The difference quotient = 0.

• The function f(x) = x² - 7x + 5

f(x + h) = (x + h)² - 7(x + h) + 5

            = x² + h² + 2xh - 7x - 7h + 5

f(x + h) - f(x) = (x² + h² + 2xh - 7x - 7h + 5) - (x² - 7x + 5)

                    = x² + h² + 2xh - 7x - 7h + 5 - x² + 7x - 5

                   = h² + 2xh - 7h

The difference quotient = (h² + 2xh - 7h)/h

                                      = h(h + 2x - 7)/h

                                      = h + 2x - 7

The difference quotient = h + 2x - 7.