In Exercises 41–44:
a. Find f⁻¹(x).

43. f(x) = 5 − 4x, a = 1/2

검증된 단계별 안내

Start by writing the function given: \(f(x) = 5 - 4x\).

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = 5 - 4x\).

Swap the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = 5 - 4y\).

Solve this equation for \(y\) to express \(y\) in terms of \(x\): subtract 5 from both sides to get \(x - 5 = -4y\), then divide both sides by \(-4\) to isolate \(y\).

Write the inverse function as \(f^{-1}(x) = \frac{5 - x}{4}\) (after simplifying the expression).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

54s

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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. If f(x) maps x to y, then f⁻¹(x) maps y back to x. Finding the inverse involves solving the equation y = f(x) for x in terms of y.

Solving Linear Equations

Since f(x) = 5 − 4x is linear, finding its inverse requires isolating x in the equation y = 5 − 4x. This involves algebraic manipulation such as adding, subtracting, multiplying, or dividing both sides to solve for x.

Domain and Range Considerations

When finding an inverse, it is important to consider the domain and range of the original function and its inverse. The value a = 1/2 may specify a particular input or output to check, ensuring the inverse function is valid at that point.

In Exercises 41–44:a. Find f⁻¹(x).43. f(x) = 5 − 4x, a = 1/2