Carcass wrote:
If f(x)=x−1 and g(x)=x2−1, which of the following pairs of x-values satisfy the equation g(f(x))=0?
(A) -2, 1
(B) -1, 0
(C) 0, 1
(D) 0, 2
(E) 1, 2
Approach #1: AlgebraWe want values of x such that
g(f(x))=0Let
f(x)=kSo we want:
g(f(x))=g(k)=0Plug
k into the function
g to get:
g(k)=k2−1=0Add
1 to both sides of the equation:
k2=1So, EITHER
k=1 OR
k=−1 In other words, EITHER
f(x)=1 OR
f(x)=−1 Since
f(x)=x−1, we now have two equations to solve:
If
f(x)=1, then we have:
x−1=1, which means
x=2 is one possible solution.
If
f(x)=−1, then we have:
x−1=−1, which means
x=0 is another possible solution.
Answer: D
Approach #2: Test values from the answer choices (this very well may be the faster approach)
Let's first see if
x=0 is a solution.
f(0)=0−1=−1.
So,
g(f(0))=g(−1)=(−1)2−1=1−1=0. WORKS.
Since
x=0 is a solution, we can eliminate answer choices A and E, since they don't include
x=0 as a solution
Now let's see if
x=1 is a solution.
f(1)=1−1=0.
So,
g(f(1))=g(0)=(0)2−1=0−1=−1. Doesn't work.
Since
x=1 is NOT a solution, we can eliminate answer choice C, since it says
x=1 is a solution
Now let's see if
x=2 is a solution.
f(2)=2−1=1.
So,
g(f(2))=g(1)=(1)2−1=1−1=0. WORKS.
Answer: D