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

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

All textbooks

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.39

Chapter 1, Problem 1.39

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

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

4:49

Inverse Cosine

Related Practice

Textbook Question

Solving equations Solve each equation.

√2 sin 3Θ + 1 = 2, 0 ≤ Θ ≤ π

Textbook Question

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

$the log base 6 of 60$

views

Textbook Question

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

views

Textbook Question

Evaluate each expression without a calculator.

a. log₁₀ 1000

views

Textbook Question

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4