Composing Functions at a Point

The composition of f(x) with g(x) is written as "(f ∘ g)(x)", which is pronounced as "f-compose-g of x". And "(f ∘ g)(x)" means "f(g(x))", which is "f-of-g of x".

When composing functions f and g, you plug something in for x, then you plug that value into g, simplify, and then plug the result into f. The process here is just like what we saw on the previous page, except that now we will be using formulas to find the output values, rather than just reading the values from lists of points or from graphs.

What is an example of composing two functions and evaluating at a given x-value?

• Given f (x) = 2x + 3 and g(x) = −x² + 5, find (g ∘ f )(1).

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Above, I worked with the composition (g ∘ f )(1); the composition can also work with the functions in the opposite order:

• Given f (x) = 2x + 3 and g(x) = −x² + 5, find (f ∘ g)(1).

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What are "fog" and "gof"?

These are mispronunciations based on treating the circle character, ∘, as a lowercase letter O: despite what some might say, "f ∘ g" is not pronounced as "fogg" and "g ∘ f " is not pronounced as "goff". They are pronounced as "f-compose-g" and "g-compose-f", respectively.

Don't make yourself sound ignorant by pronouncing these wrongly.

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As you have seen above, you can plug one function into another. You can also plug a function into itself.

What is an example of a function composed with itself?

• Given f (x) = 2x + 3 and g(x) = −x² + 5, find (f ∘ f )(1).

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• Given f (x) = 2x + 3 and g(x) = −x² + 5, find (g ∘ g)(1).

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In each of these exercises, using either method of doing the computations (whether bit-by-bit or all in one go), I wrote out my steps carefully, and used parentheses to indicate where my input went with respect to a function's formula. You should use the method — bit-by-bit or all at once — that works better for you. Whichever method you pick, though, take the time to be clear in each of your steps.