In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

검증된 단계별 안내

Start with the given function: \(y = \sqrt{3x - 2}\), where \(\frac{2}{3} \leq x \leq 4\).

To find the inverse function, first express the equation in terms of \(x\): square both sides to eliminate the square root, giving \(y^2 = 3x - 2\).

Next, solve for \(x\) by isolating it on one side: add 2 to both sides to get \(y^2 + 2 = 3x\), then divide both sides by 3 to obtain \(x = \frac{y^2 + 2}{3}\).

Define the inverse function \(g\) as \(g(y) = \frac{y^2 + 2}{3}\), which expresses \(x\) as a function of \(y\).

Verify the domain and range of \(g\) to ensure it matches the original function's range and domain, considering the original domain \(\frac{2}{3} \leq x \leq 4\).

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.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse g(y) satisfies g(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y, ensuring the function is one-to-one on the given domain.

Domain and Range Restrictions

To have an inverse, a function must be one-to-one, often requiring domain restrictions. The given interval 2/3 ≤ x ≤ 4 ensures the function is invertible by limiting its domain. Understanding these restrictions is crucial to correctly define and find the inverse function.

Derivative and Tangent Line Approximation

The derivative of a function at a point gives the slope of the tangent line there, which can approximate the function near that point. For inverse functions, the derivative of the inverse at a point relates to the reciprocal of the original function's derivative, aiding in tangent line calculations.