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.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

검증된 단계별 안내

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

To find the inverse function \(g\), solve the equation \(y = 2 - x - x^{3}\) for \(x\) in terms of \(y\).

Rewrite the equation as \(x^{3} + x = 2 - y\) to isolate terms involving \(x\) on one side.

Recognize that this is a cubic equation in \(x\): \(x^{3} + x - (2 - y) = 0\).

Use an appropriate method (such as the cubic formula or a computer algebra system) to solve for \(x\) as a function of \(y\), and define this solution as the inverse function \(g(y)\).

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, which may require algebraic manipulation and checking domain restrictions to ensure the inverse is well-defined.

Derivative of Inverse Functions

The derivative of an inverse function at a point relates to the derivative of the original function by the formula g'(y) = 1 / f'(x), where y = f(x). This relationship helps find the slope of the tangent line to the inverse function without explicitly differentiating it, provided f'(x) ≠ 0 at the point of interest.

Tangent Line Approximation

Tangent line approximation uses the derivative at a point to approximate the function near that point with a linear function. For a function f at x₀, the tangent line is y = f(x₀) + f'(x₀)(x - x₀). This concept extends to inverse functions, allowing approximation of g(y) near y₀ using the slope of the inverse function.