Find the inverse f−$$1(x)f^{-1}$$\left(x\right)f−1(x) of each function (on the given interval, if specified).f(x)=10−2xf\left(x\right)=$$10^{-2x}f(x)=10$$−2x

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problema 1.39

Capitolo 1, Problema 1.39

Find the inverse $f^{-1}$\left$(x$\right$)$ of each function (on the given interval, if specified).
$f$\left$(x$\right$)=10^{-2x}$

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Start by replacing f(x) with y to make the equation easier to work with: y = 10 - 2x.

To find the inverse, swap x and y in the equation. This gives us: x = 10 - 2y.

Solve for y in terms of x. Begin by isolating the term with y: subtract 10 from both sides to get x - 10 = -2y.

Divide both sides by -2 to solve for y: y = (10 - x) / 2.

The inverse function is f^{-1}(x) = (10 - x) / 2. This is the expression for the inverse of the given function.

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Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f^{-1} takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

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Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where a is a constant, b is the base (a positive real number), and x is the exponent. In the given function f(x) = 10^{-2x}, the base is 10, and the exponent is -2x, indicating that the function decreases rapidly as x increases, which is characteristic of exponential decay.

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Exponential Functions

Finding Inverses Algebraically

To find the inverse of a function algebraically, you typically start by replacing f(x) with y, then solve for x in terms of y. After isolating x, you swap x and y to express the inverse function. This process often involves manipulating equations, such as applying logarithms to exponential functions, to derive the inverse correctly.

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Inverse Cosine

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