Verifying if Two Functions are Inverses of Each Other

Previously, you learned how to find the inverse of a function. This time, you will be given two functions and will be asked to prove or verify if they are inverses of each other. But how?

The procedure is really simple. But before I do so, I want you to get some basic understanding of how the “verifying” process works. Let’s take a look at the diagram below.

Suppose you’re given two functions, namely, [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex].

Think about this, if I plug [latex]x=2[/latex] into [latex]f\left( x \right)[/latex], I get an output of [latex]\color{blue}5[/latex]. Now, I plug [latex]x=5[/latex] into [latex]g\left( x \right)[/latex]. What is the output value?

Well, I got the answer [latex]\color{red}2[/latex] which is the original input value for [latex]f\left( x \right)[/latex] of the function I started with. That’s a good observation. Can I say, what goes around comes around? This is precisely the main idea of what we’re going to do when we are asked to verify or prove if two functions are inverses of each other.

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Steps on How to Verify if Two Functions are Inverses of Each Other

Verifying if two functions are inverses of each other is a simple two-step process.

STEP 1:

• Plug [latex]g\left( x \right)[/latex] into [latex]f\left( x \right)[/latex], then simplify.

• If true, move to Step 2.
• If false, STOP! That means [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are not inverses.

STEP 2:

• Plug [latex]f\left( x \right)[/latex] into [latex]g\left( x \right)[/latex], then simplify.

• If true again, then [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are inverses. Success!
• If false, then [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are not inverses.

Technically, for [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] to be inverses of each other, you must show that function composition works both ways! Therefore, the composition of function [latex]\color{blue}f[/latex] with [latex]\color{red}g[/latex] equals [latex]x[/latex], and vice versa. It is “elegantly” summarized in the equation below.

CONCLUSION:

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Examples of How to Verify if Two Functions are Inverses of Each Other

Example 1: Verify or prove the functions are inverses of each other.

• For Step 1, I will take [latex]g\left( x \right)[/latex] as the input of [latex]f\left( x \right)[/latex]. That means I will substitute whatever the equation of [latex]g\left( x \right)[/latex] to every [latex]x[/latex] in [latex]f\left( x \right)[/latex], then simplify.

• For Step 2, I simply reverse the process, that is, make [latex]f\left( x \right)[/latex] as the input of [latex]g\left( x \right)[/latex].

Since the results above came out very nicely, both [latex]x[/latex], then I can claim that functions [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are indeed inverses of each other.

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Example 2: Verify or prove the functions are inverses of each other.

I have two rational functions, and so I expect the composition process to be a bit tedious either way. As long as I am being careful and actually making small progress in every step of my solution, then it should encourage me to finally get it right!

I start composing [latex]g\left( x \right)[/latex] with [latex]f\left( x \right)[/latex].

The simplified answer is [latex]2x[/latex] which missed our target of just [latex]\large\color{green}x[/latex]. I would stop here right away and conclude that [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are NOT inverses of each other.

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Example 3: Verify or prove the functions are inverses of each other.

• Show that [latex]f\left[ {g\left( x \right)} \right] = x[/latex]

• Show that [latex]g\left[ {f\left( x \right)} \right] = x[/latex].

Since both outputs are [latex]\large\color{green}x[/latex] then [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are inverses of each other!

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Example 4: Verify or prove the functions are inverses of each other.

I will substitute the formula of [latex]g\left( x \right)[/latex] into [latex]f\left( x \right)[/latex], then simplify.

Remember we need to arrive at [latex]\large\color{green}x[/latex], that is, [latex]x[/latex] with a coefficient of [latex]+1[/latex], and not [latex]-1[/latex]. What we have though is a coefficient of [latex]-1[/latex] which is NOT what we want. Therefore, this slight or small difference should make us conclude that [latex]f\left( x \right)[/latex] and [latex]g\left( x \right)[/latex] are NOT inverses of each other which means that there’s no need to do Step 2.