1.8 Graphs of Inverse Functions

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Main Content

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1. What Is the Line y = x?

Before we look at inverse function graphs, you need to be comfortable with the line y = x.

• This is a straight diagonal line that passes through the origin (0, 0)
• Every point on this line has an equal x-coordinate and y-coordinate — for example: (1, 1), (2, 2), (3, 3), (−4, −4)
• It goes diagonally at a 45° angle, slanting upward to the right
• This line acts like a mirror in our topic — it is the mirror line for inverse function graphs

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2. The Relationship Between a Function and Its Inverse — The Reflection Rule

This is the core idea of this entire subtopic, so read carefully.

When you draw the graph of a function f(x) and its inverse f⁻¹(x) on the same set of axes, the two graphs are mirror images (reflections) of each other in the line y = x.

• The graph of f⁻¹(x) is obtained by reflecting the graph of f(x) in the line y = x
• This works the other way too — reflecting f⁻¹(x) in the line y = x gives you back f(x)
• This is true for all one-to-one functions and their inverses

Example to See This Clearly

Consider f(x) = 2x − 1 for the domain −1 ≤ x ≤ 3.

• The domain of f is: −1 ≤ x ≤ 3 (the x-values the function accepts)

• The range of f is: −3 ≤ f(x) ≤ 5 (the y-values the function produces)

  • When x = −1: f(−1) = 2(−1) − 1 = −3 → point (−1, −3)
  • When x = 3: f(3) = 2(3) − 1 = 5 → point (3, 5)

The inverse function is: f⁻¹(x) = (x + 1) ÷ 2

• The domain of f⁻¹ is: −3 ≤ x ≤ 5 (the range of f becomes the domain of f⁻¹)

• The range of f⁻¹ is: −1 ≤ f⁻¹(x) ≤ 3 (the domain of f becomes the range of f⁻¹)

  • When x = −3: f⁻¹(−3) = (−3 + 1) ÷ 2 = −1 → point (−3, −1)
  • When x = 5: f⁻¹(5) = (5 + 1) ÷ 2 = 3 → point (5, 3)

Now notice something important:

The coordinates are swapped — the x and y values switch. This is exactly what a reflection in y = x does.