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 ) = 1 0 − 2 x f\left(x\right)=10^{-2x} f ( x ) = 1 0 − 2 x

Welcome back, everyone. Find the inverse function G inverse of X of G of X equals five to the power of negative three X were given for answer choices A states three log X divided by log five B states five log X divided by log three C negative log X divided by five log three and D negative log X divided by three log five. So first of all, let's define our function and let's replace G of X with Y because that's how Y coordinate. So Y equals five to the power of negative three X. We want to identify the inverse function meaning we have to switch X and Y so X equals five to the power of negative three Y. And since we're solving for the inverse function we want to solve or why? Since Y is part of the exponent, we're going to take the log of both sides because logarithms allow us to rewrite exponents as leading coefficients. What type of log are we going to take? Well, we're going to take the log of base 10 because those are the answer choices that use a log of base 10, we don't see any base, meaning by definition, the base is 10. So if we take a log of both sides, then log of X must be equal to log of five to the power of negative three Y. And from here, let's recall that we can bring down the exponent based on the properties of logs such that log of eight, the power of X is simply X multiplied by a log of A. So a lot of acts must be equal two. Our exponent is negative three Y and we end up with lock five. Now solving for Y, we have to divide both sides by negative log of five. So essentially Y equals the left hand side, log X divided by the right hand side. So we have negative, we log five. So we end up with a negative sign, negative sign and then we lock five in the denominator. So now if we look at the answer choices, the final answer would be negative log X divided by three log five, which is option D of the multiple choice. Thank you for watching.

0. Functions

Common Functions

Calculus

0. Functions

Common Functions

2:46 minutes

Problem 1.39

Textbook Question

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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