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Math  /  Algebra

Questionf−1(x)=x−12f^{-1}(x)=\sqrt{\frac{x-1}{2}} is the inverse of f(x)=2x2−1f(x)=2 x^{2}-1. a) True b) False

Studdy Solution

STEP 1

1. We are given the inverse function f−1(x)=x−12 f^{-1}(x) = \sqrt{\frac{x-1}{2}} .
2. We need to verify if f(x)=2x2−1 f(x) = 2x^2 - 1 is the function whose inverse is given.
3. For f−1(x) f^{-1}(x) to be the correct inverse, f(f−1(x)) f(f^{-1}(x)) should equal x x and f−1(f(x)) f^{-1}(f(x)) should equal x x .

STEP 2

1. Verify if applying f f to f−1(x) f^{-1}(x) yields x x .
2. Verify if applying f−1 f^{-1} to f(x) f(x) yields x x .

STEP 3

Define the function and its claimed inverse:
f(x)=2x2−1andf−1(x)=x−12f(x) = 2x^2 - 1 \quad \text{and} \quad f^{-1}(x) = \sqrt{\frac{x-1}{2}}

STEP 4

Apply the function f f to its claimed inverse f−1(x) f^{-1}(x) :
f(f−1(x))=f(x−12)f(f^{-1}(x)) = f\left(\sqrt{\frac{x-1}{2}}\right)

STEP 5

Substitute x−12 \sqrt{\frac{x-1}{2}} into f(x) f(x) :
f(x−12)=2(x−12)2−1f\left(\sqrt{\frac{x-1}{2}}\right) = 2\left(\sqrt{\frac{x-1}{2}}\right)^2 - 1

STEP 6

Simplify the expression:
2(x−12)−1=x−12\left(\frac{x-1}{2}\right) - 1 = x - 1

STEP 7

Verify if the result is x x :
x−1+1=xx - 1 + 1 = x
This confirms that f(f−1(x))=x f(f^{-1}(x)) = x .

STEP 8

Apply the inverse function f−1 f^{-1} to f(x) f(x) :
f−1(f(x))=f−1(2x2−1)f^{-1}(f(x)) = f^{-1}(2x^2 - 1)

STEP 9

Substitute 2x2−1 2x^2 - 1 into f−1(x) f^{-1}(x) :
f−1(2x2−1)=2x2−1−12f^{-1}(2x^2 - 1) = \sqrt{\frac{2x^2 - 1 - 1}{2}}

STEP 10

Simplify the expression:
2x2−22=x2=∣x∣\sqrt{\frac{2x^2 - 2}{2}} = \sqrt{x^2} = |x|

STEP 11

Verify if the result is x x :
Since the inverse function should return x x itself, and we obtained ∣x∣ |x| , this indicates that f−1(f(x))=∣x∣ f^{-1}(f(x)) = |x| .
Given the absolute value, f−1(x)=x−12 f^{-1}(x) = \sqrt{\frac{x-1}{2}} is not the true inverse of f(x)=2x2−1 f(x) = 2x^2 - 1 for all x x .
Therefore, the statement is:
False\boxed{\text{False}}

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