Composite and inverse functions

Objective. Combine and reverse functions with domain restrictions.

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Learn: the key idea

Composite and inverse functions combine or reverse the action of two functions: fg(x) means apply g first then f, written f(g(x)), while the inverse f⁻¹(x) undoes whatever f(x) does, found by swapping x and y in y=f(x) and rearranging for y. Domain restrictions matter because the range of the inner function must lie within the domain of the outer function for a composite to be valid, and an inverse only exists properly where the original function is one-to-one. To find fg(x), substitute the entire expression for g(x) wherever x appears in f. For example, if f(x)=2x+1 and g(x)=x², then fg(x)=2x²+1, but gf(x)=(2x+1)².

Worked example

Given f(x)=3x−2, find f⁻¹(x), the inverse function.

1 $Write y = 3x-2$
2 $Swap x and y: x = 3y-2$
3 $Rearrange for y: x+2 = 3y, so y = (x+2)/3$
4 $Write the inverse using function notation$

Answer: f⁻¹(x) = (x+2)/3

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Common mistake. Students often confuse the order of composite functions, calculating fg(x) as g(f(x)) instead of f(g(x)); remember fg(x) means apply the function closest to x first, so g acts on x before f acts on the result.

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