Textbook Question
Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2^x-1
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6. Exponential & Logarithmic Functions
Solving Exponential and Logarithmic Equations
3:03 minutesProblem 112
Textbook Question
Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)^x
Verified step by step guidance1
Start by replacing \( f(x) \) with \( y \) to get \( y = \left( \frac{1}{3} \right)^x \).
Swap \( x \) and \( y \) to find the inverse: \( x = \left( \frac{1}{3} \right)^y \).
Take the logarithm of both sides to solve for \( y \): \( \log(x) = \log\left( \left( \frac{1}{3} \right)^y \right) \).
Use the logarithmic identity \( \log(a^b) = b \cdot \log(a) \) to rewrite the equation: \( \log(x) = y \cdot \log\left( \frac{1}{3} \right) \).
Solve for \( y \) by dividing both sides by \( \log\left( \frac{1}{3} \right) \): \( y = \frac{\log(x)}{\log\left( \frac{1}{3} \right)} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A one-to-one function is a type of function where each output is produced by exactly one input. This means that no two different inputs can yield the same output. This property is crucial for finding the inverse of a function, as only one-to-one functions have inverses that are also functions.
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Inverse Functions
An inverse function essentially reverses the effect of the original function. If a function ƒ takes an input x to produce an output y, then its inverse function ƒ⁻¹ takes y back to x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y.
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Exponential Functions
Exponential functions are mathematical expressions in the form ƒ(x) = a^x, where a is a positive constant. The function given, ƒ(x) = (1/3)^x, is an exponential function with a base of 1/3. Understanding the properties of exponential functions is essential for determining their inverses, which are logarithmic functions.
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