for f(x)=(x-6)/(x+9) construct and simplify the difference quotient
Solution
To construct and simplify the difference quotient for the function f(x)=x+9x−6, we follow these steps:
Difference Quotient Formula: The difference quotient for a function f(x) is given by:
hf(x+h)−f(x)
Substitute f(x) and f(x+h) into the formula:
f(x)=x+9x−6f(x+h)=(x+h)+9(x+h)−6=x+h+9x+h−6
Construct the difference quotient:
hf(x+h)−f(x)=hx+h+9x+h−6−x+9x−6
Simplify the numerator:
x+h+9x+h−6−x+9x−6
To combine these fractions, find a common denominator:
(x+h+9)(x+9)(x+h−6)(x+9)−(x−6)(x+h+9)
Expand and combine the terms in the numerator:
(x+h+9)(x+9)(x+h−6)(x+9)−(x−6)(x+h+9)=(x+h+9)(x+9)(x2+9x+hx+9h−6x−54)−(x2+hx+9x+9h−6x−54)
Simplifying the expanded terms:
=(x+h+9)(x+9)(x2+9x+hx+9h−6x−54)−(x2+hx+9x+9h−6x−54)
Notice that all the terms cancel each other out in the numerator:
=(x+h+9)(x+9)(x2+9x+hx+9h−6x−54)−(x2+hx+9x+9h−6x−54)=(x+h+9)(x+9)0
This leads to:
=h(x+h+9)(x+9)h(x−6)(x+9)−h(x−6)(x+h+9)
Factor out h in the numerator:
=h(x+h+9)(x+9)h[(x−6)(x+9)−(x−6)(x+h+9)]
Simplify the numerator further:
=h(x+h+9)(x+9)h[(x−6)(x+9)−(x−6)(x+h+9)]=h(x+h+9)(x+9)h(x−6)(x+9−x−h−9)=h(x+h+9)(x+9)h(x−6)(−h)=h(x+h+9)(x+9)−h(x−6)
Cancel the h term:
=(x+h+9)(x+9)−(x−6)
Thus, the simplified difference quotient for f(x)=x+9x−6 is:
(x+h+9)(x+9)−(x−6)
Would you like any further details or have any questions? Here are some related questions you might find helpful:
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Tip: When simplifying the difference quotient, be meticulous with algebraic manipulations to avoid errors in combining and canceling terms.