Inverse Function Graphs

Grade 12 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

🧭 What is an Inverse Function Graph?

An inverse function essentially "reverses" the action of the original function. If a function f maps an input x to an output y, then its inverse, f⁻¹, maps y back to x. Graphically, the inverse is the reflection of the original function's graph across the line y = x. This is crucial for solving equations where the variable is trapped inside a function.

📝 Step-by-Step Guide: Graphing an Inverse

  1. Check if the function is one-to-one. It must pass the Horizontal Line Test to have an inverse.
  2. Graph the original function. Plot key points like the y-intercept, x-intercept, and a few others.
  3. Reflect across the line y = x. For every point (a, b) on the original graph, plot a new point (b, a) for the inverse.
  4. Draw the line of reflection. Lightly sketch the line y = x to guide your reflection.
  5. Sketch the inverse graph. Connect the new reflected points smoothly.

🔍 Visual Examples

Example 1: Linear Function

Function: f(x) = 2x - 4

Step 1: Original graph has points like (0, -4), (2, 0).

Step 2: Reflect points to get (-4, 0), (0, 2).

Result: The inverse graph is a line through these new points. Its equation is f⁻¹(x) = (x + 4)/2.

Example 2: Restricted Quadratic Function

Function: f(x) = x², for x ≥ 0

Step 1: Original graph is the right half of a parabola with points (0,0), (1,1), (4,2).

Step 2: Reflect points to get (0,0), (1,1), (2,4).

Result: The inverse graph is f⁻¹(x) = √x, which is the square root function.

⚠️ Common Mistakes

Forgetting the Horizontal Line Test: Not every function has an inverse. If a function isn't one-to-one (like f(x) = x² for all real numbers), you must restrict its domain first.

Mis-Reflecting Points: Students often mix up the x and y coordinates when reflecting. Remember: (a, b) reflects to (b, a).

Confusing Notation: f⁻¹(x) does NOT mean 1/f(x). This is a common notation error.

💡 Tips & Tricks

  • The "Switcheroo" Method: To find the equation of an inverse algebraically, switch x and y in the original equation, then solve for y.
  • Domain & Range Swap: The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. This is a great check!
  • Visual Check: The original function and its inverse should look like mirror images across the line y = x.

🎯 Practice Suggestions

To master inverse graphs:

  1. Start with simple linear functions to get the reflection technique down.
  2. Practice with basic non-linear functions like f(x) = x³ and f(x) = √x.
  3. Use graph paper or digital tools (Desmos, GeoGebra) to plot a function and its inverse simultaneously to see the reflection.
  4. Always state the domain restrictions for functions that aren't naturally one-to-one.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 The graph of function f shows f(4) = 8. Find f⁻¹(8).

Hint: Remember that for inverse functions, the input and output values are swapped. If f(a) = b, then f⁻¹(b) = a.

Show the answer

Answer: 4

  1. The problem states that f(4) = 8, meaning when x = 4, f(x) = 8.
  2. For inverse functions, the coordinates are swapped. If (4, 8) is a point on f, then (8, 4) is a point on f⁻¹.
  3. Therefore, f⁻¹(8) = 4.

The answer is 4.

2 The graph of function f shows f(5)=9. What is f⁻¹(9)?

Hint: For inverse functions, the input and output values swap positions. If f(a)=b, then f⁻¹(b)=a.

Show the answer

Answer: 5

  1. The given information is f(5)=9, which means when x=5, f(x)=9.
  2. For inverse functions, the coordinates swap: if (a,b) is on the graph of f, then (b,a) is on the graph of f⁻¹.
  3. Since f(5)=9, the point (5,9) is on the graph of f.
  4. Therefore, the point (9,5) is on the graph of f⁻¹.
  5. This means f⁻¹(9)=5.

The answer is 5.

3 The graph of function f shows f(4)=8. What is f⁻¹(8)?

Hint: Remember that for inverse functions, the input and output values are swapped. If a point (a,b) is on the graph of f, then (b,a) is on the graph of f⁻¹.

Show the answer

Answer: 4

  1. The graph shows f(4) = 8, which means the point (4,8) is on the graph of f.
  2. For inverse functions, the coordinates are swapped. So if (4,8) is on f, then (8,4) is on f⁻¹.
  3. This means f⁻¹(8) = 4.

The answer is 4.

4 The graph of function f shows f(7) = 2. What is f⁻¹(2)?

Hint: Remember that for inverse functions, if f(a) = b, then f⁻¹(b) = a. Think about swapping the input and output values.

Show the answer

Answer: 7

  1. The problem states that f(7) = 2, meaning when the input is 7, the output is 2.
  2. For the inverse function f⁻¹, the input and output values are swapped.
  3. Therefore, if f(7) = 2, then f⁻¹(2) = 7.
  4. The answer is 7.

5 The graph of function f shows f(8)=11. What is f⁻¹(11)?

Hint: Remember that for inverse functions, the input and output values swap positions. If a point (a,b) is on the graph of f, then (b,a) is on the graph of f⁻¹.

Show the answer

Answer: 8

  1. The given information is f(8) = 11, which means when x = 8, f(x) = 11.
  2. For inverse functions, f⁻¹(b) = a when f(a) = b.
  3. Since f(8) = 11, we know f⁻¹(11) = 8.
  4. The answer is 8.

6 The graph of function f shows f(5)=10. What is f⁻¹(10)?

Hint: For inverse functions, the input and output values are swapped. If a point (a,b) is on the original function, then (b,a) is on the inverse function.

Show the answer

Answer: 5

  1. The given information is f(5) = 10, which means when x = 5, f(x) = 10.
  2. For inverse functions, the coordinates are swapped. So if (5,10) is on the graph of f, then (10,5) is on the graph of f⁻¹.
  3. This means f⁻¹(10) = 5.

The answer is 5.

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