The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^-1(x), the inverse function. (b) Verify that your equation is correct by showing that f(f^-1(x)) = x and f^-1(f(x)) = x. f(x) = 4x - 3

검증된 단계별 안내

Start with the given function: \(f(x) = 4x - 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = 4x - 3\).

Next, swap the roles of \(x\) and \(y\) to find the inverse: replace \(y\) with \(x\) and \(x\) with \(y\), giving \(x = 4y - 3\).

Solve this new equation for \(y\) to express the inverse function: add 3 to both sides to get \(x + 3 = 4y\), then divide both sides by 4 to isolate \(y\): \(y = \frac{x + 3}{4}\).

Rewrite \(y\) as \(f^{-1}(x)\), so the inverse function is \(f^{-1}(x) = \frac{x + 3}{4}\).

To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f\): \(f\left(f^{-1}(x)\right) = 4 \times \frac{x + 3}{4} - 3\). Simplify this expression and check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}\): \(f^{-1}(4x - 3) = \frac{(4x - 3) + 3}{4}\). Simplify and check if it equals \(x\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, meaning it passes the horizontal line test. This property ensures the function has an inverse because no two different inputs produce the same output.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, denoted as f⁻¹(x). To find the inverse, solve the equation y = f(x) for x in terms of y, then interchange x and y. The inverse 'undoes' the original function.

Verification of Inverse Functions

To verify that two functions are inverses, show that composing them in either order returns the input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that each function reverses the effect of the other.

The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f-1(x), the inverse function. (b) Verify that your equation is correct by showing that f(f-1(x)) = x and f-1(f(x)) = x. f(x) = 4x - 3