Given a linear function: \[f(x) = ax+b\] we now learn about how to find its inverse function: \[f^{-1}(x)\]
We've see what an inverse function is and we've seen that a function \(f(x)\) only has an inverse if it is a one-to-one mapping.
A linear functions, \(f(x) = ax+b\) is represented by a line with equation \(y = ax+b\), which passes the horizontal line test and is definitely a one-to-one mapping; linear functions therefore have an inverse.
The content of this page, and what we'll learn here, can be summarized as follows:
The method for finding a function's inverse can be summarized in two steps:
In this tutorial, we show how to use our two-step method for finding the inverse function of a linear function, \(f(x) = ax+b\).
In particular we find the inverse function of the following two functions: \[f(x) = 3x \quad \text{and} \quad f(x) = 2x+8\]
Given the function defined by: \[f(x) = 2x+4\] find an expression for its inverse function \(f^{-1}(x)\).
In this tutorial, we show how to use our two-step method for finding the inverse function of a linear function, \(f(x) = \frac{x}{a}+b\).
In particular we find the inverse function of the following two functions: \[f(x) = \frac{x}{3} \quad \text{and} \quad f(x) = \frac{x}{5} + 1 \]
Find the inverse function for each of the following functions:
We now learn how to find the inverse of a rational function, of the form: \[f(x) = \frac{ax+b}{cx+d}\] For example, we'll know how to find the inverse function of: \[f(x) = \frac{2x+5}{x-3}\]
Find the inverse function for each of the following functions:
At times finding the inverse function, \(f^{-1}(x)\), won't be quite as obvious.
In exams we'll often be asked to find the inverse function of a quadratic function, for which we're told \(x\geq p\) or \(x\leq q\), where \(p\) and \(q\) could be any two real numbers.
Say we're given the function \(f(x)=x^2\), for \(x\geq 0\), and we're asked to find \(f^{-1}(x)\).
Following our twp-step method for finding the inverse leads to:
Find the inverse function