Simplify the difference quotient


[f(x) − f(a]/[x − a]

if x ≠ a.
f(x) = SQROOT of (x − 27)
(Hint: Rationalize the numerator.)

y(x) = (x-27)^.5

y(a) = (a -27)^.5

y(x) -y(a) = (x-27)^.5 - (a-27)^.5)
then multiply top and bottom by
(x-27)^.5 + (a-27)^.5)
and get
(x-27)-(a-27)/{(x-a)[(x-27)^.5 - (a-27)^.5)]}

= (x-a)/{(x-a)[(x-27)^.5 - (a-27)^.5)]}

= 1/[(x-27)^.5 - (a-27)^.5)]

To simplify the difference quotient [f(x) - f(a)] / [x - a], we need to substitute the given expression for f(x) and simplify.

Given:
f(x) = √(x - 27)

Let's start by substituting f(x) and f(a) into the difference quotient:

[f(x) - f(a)] / [x - a] = [√(x - 27) - √(a - 27)] / [x - a]

To rationalize the numerator, we need to eliminate the square root. We can do this by multiplying the numerator and denominator by the conjugate of the numerator:

[√(x - 27) - √(a - 27)] / [x - a] * [√(x - 27) + √(a - 27)] / [√(x - 27) + √(a - 27)]

Multiplying the numerators and denominators together, we get:

[√(x - 27) * √(x - 27) - √(x - 27) * √(a - 27) - √(a - 27) * √(x - 27) + √(a - 27) * √(a - 27)] / [(x - a) * (√(x - 27) + √(a - 27))]

Simplifying further:

[(x - 27) - √[(x - 27) * (a - 27)] - √[(x - 27) * (a - 27)] + (a - 27)] / [(x - a) * (√(x - 27) + √(a - 27))]

Combining like terms:

[x - 27 - 2√[(x - 27) * (a - 27)] + a - 27] / [(x - a) * (√(x - 27) + √(a - 27))]

Now we can simplify further:

[x + a - 54 - 2√[(x - 27) * (a - 27)]] / [(x - a) * (√(x - 27) + √(a - 27))]

Thus, the simplified difference quotient is:

[x + a - 54 - 2√[(x - 27) * (a - 27)]] / [(x - a) * (√(x - 27) + √(a - 27))]

To simplify the given difference quotient [f(x) − f(a)] / [x − a], we need to substitute the given function f(x) = √(x − 27) and apply the hint of rationalizing the numerator.

Let's substitute the function into the difference quotient expression:

[√(x − 27) − √(a − 27)] / [x − a]

To rationalize the numerator, we multiply both the numerator and the denominator by the conjugate of the numerator, which is √(x − 27) + √(a − 27). This will eliminate the square root in the numerator:

{[√(x − 27) − √(a − 27)] * [√(x − 27) + √(a − 27)]} / [x − a]
= [(x − 27) − (a − 27)] / [x − a]

Now we can simplify further:

= (x − 27 − a + 27) / (x − a)
= (x - a) / (x - a)

Finally, we can see that the numerator and denominator are the same, which means they cancel out:

= 1

Therefore, the simplified difference quotient is 1 when x ≠ a.