Question

f(x +h)-f(x)Find the difference quotient off, that is find,h70, for the function f(x) = √ X-15. (Hint: Rationalize the numerator.]hThe difference quotient off, f(x) = √ X-15 is(Simplify your answer.)

Answer 1

We have:

[f(x + h) - f(x)]/h

Where h ≠ 0, and the function f is:

f(x) = √(x - 15)

Now, we can calculate f(x + h):

f(x + h) = √((x + h) - 15) = √(x + h - 15)

Then:

DQ = [f(x + h) - f(x)]/h = [√(x + h - 15) - √(x - 15)]/h

To rationalize the numerator, we multiply and divide the right side by

[√(x + h - 15) + √(x - 15)], because we know that:

(a + b) * (a - b) = a² - b²

Here:

a = √(x + h - 15)

b = √(x - 15)

Then:

[√(x + h - 15) - √(x - 15)]*[√(x + h - 15) + √(x - 15)] = x + h - 15 - x + 15 = h ...(1)

In DQ:

DQ = [√(x + h - 15) - √(x - 15)]/h*([√(x + h - 15) + √(x - 15)]/[√(x+h-15) + √(x - 15)])

*As it is shown below*

Using (1):

DQ = h/(h*[√(x + h - 15) + √(x - 15)])

Canceling the h term:

DQ = 1/([√(x + h - 15) + √(x - 15)])