Inverse Functions In Algebra
If you need some help understanding inverse functions in Algebra, this guide I wrote explains them with several examples.
Inverse Functions In Algebra
Many types of calculations are reversible in math. They take a slightly different mindset to comprehend, but are otherwise not difficult. In Algebra, we call these operations inverse functions. In essence, an inverse function is doing the actions in reverse. Inverse operations can be described by functions. If input “x” produces output “y”, then input “y” produces output “x” as the inverse function. So, outputs and inputs are interchanged for inverse functions.
Example 1
Given \( f(x) = x + 5 \) and \( f^{-1}(x) = x - 5 \) are applied in order.
\( (f^{-1} \circ f)(x) = f^{-1}(f(x)) \)
\( f^{-1}(x + 5) \)
\( (x + 5) - 5 \)
\( = x \)
If “a” represents a real number, then \( a^{-1} = \frac{1}{a} \).
So, \( 4^{-1} = \frac{1}{4} \).
If “f” represents a function, \( f^{-1}(x) = \frac{x}{n} \).
So, if \( f(x) = 5x \) then \( f^{-1}(x) = \frac{x}{5} \).
The inverse of exponents is slightly different.
So, if \( f(x) = x^3 \) then \( f^{-1}(x) = \sqrt[3]{x} \).
One To One Functions
Not every function has an inverse. This is critical to remember. So, we have to be able to determine if a function has an inverse. If different inputs of a function produce the same output, then an inverse function does not exist for the original function. However, if different inputs always produce different outputs, then the original function is one to one and has an inverse. So, every one to one function has an inverse function.
For example, \( f(x) = x^2 \) is not one to one because \( f(-2) = 4 \) and \( f(2) = 4 \). Therefore, \( f(x) = x^2 \) does not have an inverse function because an inverse function cannot receive 4 and produce both \( -2 \) and \( 2 \) as outputs.
To visually check if a function is one to one and has an inverse, we look at its graph. Inputs are the “x” values and outputs are the “y” values. If a line intersects a curve at more than one spot then the function is not one to one and does not have an inverse. So, to be a one to one function and have an inverse, a horizontal line must intersect with the curve once. This is called the horizontal line test.
Example 2
Find the inverse of \( f(x) = x^3 - 2 \)
To reverse this process, we would add 2 and then take the cube root of x.
This would be: \( f^{-1}(x) = \sqrt[3]{x + 2} \)
We get this by the process of:
\[ y = x^3 - 2 \]
Add 2 to both sides.
\[ y + 2 = x^3 \]
Take the cube root of both sides.
\[ \sqrt[3]{y + 2} \]
Interchange x and y.
\[ f^{-1}(x) = \sqrt[3]{x + 2} \]
Example 3
Find the inverse of \( f(x) = \frac{18}{25}x + 2 \)
\[ y = \frac{18}{25}x + 2 \]
Subtract 2 from both sides.
\[ y - 2 = \frac{18}{25}x \]
Multiply \( \frac{25}{18} \) which is the reciprocal of x.
\[ \frac{25}{18}(y - 2) = x \]
Interchange x and y.
\[ f^{-1}(x) = \frac{25}{18}(x - 2) \]
Domains And Ranges
Because we exchange the x and y values when finding an inverse function, the domains and ranges are exchanged too. This means the domain of \( f \) equals the range of \( f^{-1} \). Consequently, the range of \( f \) equals the domain of \( f^{-1} \).
Graphs
If the point (3,6) lies on the graph of \( f \), then the point (6,3) must lie on the graph of \( f^{-1} \). In general, if the point (a,b) lies on the graph of \( f \), then the point (b,a) lies on the graph of \(f^{-1} \). The graph of \( f^{-1} \) is a reflection of the graph of \( f \) across the line \( y = x \).
Conclusion
A function \( f \) is one to one if different inputs always result in different outputs. If a function is one to one, it has an inverse function \( f^{-1} \). The domains of \( f \) equal the range of \( f^{-1} \). The range of \( f \) equals the domain of \( f^{-1} \).