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If f(x) = x - 1 and g(x) = x2 - 1, which of the following pairs of x-v
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06 Feb 2022, 06:15
06 Feb 2022, 06:15
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If \(f(x) = x - 1\) and \(g(x) = x^2 - 1\), 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
(A) -2, 1
(B) -1, 0
(C) 0, 1
(D) 0, 2
(E) 1, 2
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Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If f(x) = x - 1 and g(x) = x2 - 1, which of the following pairs of x-v
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06 Feb 2022, 06:56
06 Feb 2022, 06:56
1
Carcass wrote:
If \(f(x) = x - 1\) and \(g(x) = x^2 - 1\), 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
(A) -2, 1
(B) -1, 0
(C) 0, 1
(D) 0, 2
(E) 1, 2
Approach #1: Algebra
We want values of x such that \(g(f(x)) = 0\)
Let \(f(x) = k\)
So we want: \(g(f(x)) = g(k) = 0\)
Plug \(k\) into the function \(g\) to get: \(g(k) = k^2 - 1 = 0\)
Add \(1\) to both sides of the equation: \(k^2 = 1\)
So, 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
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Re: If f(x) = x - 1 and g(x) = x2 - 1, which of the following pairs of x-v
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07 Aug 2022, 09:38
07 Aug 2022, 09:38
1
Given that \(f(x) = x - 1\) and \(g(x) = x^2 - 1\) and we need to find the value of x for which \(g(f(x)) = 0\)
To find the value of g(f(x) we will first substitute the value of f(x), we get
g(f(x)) = g(x-1)
Now, in order to find the value of g(x-1), we need to compare what is inside the bracket of g(x) and g(x-1)
=> We need to put x = x-1 in \(g(x) = x^2 - 1\) to get the value of g(x-1)
=> g(x-1) = \((x-1)^2 - 1\) = \(x^2\) - 2x + 1 - 1 = \(x^2\) - 2x = 0 (given)
=> x * (x-2) = 0
=>x = 0, 2
So, Answer will be D
Hope it helps!
Watch the following video to learn the Basics of Functions and Custom Characters
To find the value of g(f(x) we will first substitute the value of f(x), we get
g(f(x)) = g(x-1)
Now, in order to find the value of g(x-1), we need to compare what is inside the bracket of g(x) and g(x-1)
=> We need to put x = x-1 in \(g(x) = x^2 - 1\) to get the value of g(x-1)
=> g(x-1) = \((x-1)^2 - 1\) = \(x^2\) - 2x + 1 - 1 = \(x^2\) - 2x = 0 (given)
=> x * (x-2) = 0
=>x = 0, 2
So, Answer will be D
Hope it helps!
Watch the following video to learn the Basics of Functions and Custom Characters
