In Exercises 41–44:
a. Find f⁻¹(x).

43. f(x) = 5 − 4x, a = 1/2

Guida verificata passo dopo passo

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).

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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.

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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.

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Linearization

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.

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