Invertible Functions

Grade 12 · algebra · 100 practice problems · read aloud

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Invertible Functions 🔁

What is an Invertible Function?

An invertible function (or one-to-one function) is a function that has an inverse function. This means each input maps to exactly one output, and each output comes from exactly one input. Invertible functions are crucial because they allow us to "reverse" a process and solve for original inputs given outputs.

How to Determine if a Function is Invertible

  1. Check if it's one-to-one: Use the horizontal line test on its graph
  2. Algebraic check: If f(a) = f(b) implies a = b, it's one-to-one
  3. Find the inverse: Replace f(x) with y, swap x and y, then solve for y
  4. Verify: Check that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Worked Examples

Example 1: f(x) = 2x + 3

1. y = 2x + 3
2. Swap: x = 2y + 3
3. Solve: x - 3 = 2y → y = (x - 3)/2
4. f⁻¹(x) = (x - 3)/2 ✅

Example 2: f(x) = x² (for x ≥ 0)

1. y = x²
2. Swap: x = y²
3. Solve: y = √x (only principal root since x ≥ 0)
4. f⁻¹(x) = √x ✅
Note: Without domain restriction, x² is not invertible!

⚠️ Common Mistakes

  • Forgetting domain restrictions: Functions like x² are only invertible on restricted domains
  • Incorrect swapping: Remember to swap x and y before solving for y
  • Assuming all functions are invertible: Always verify with the horizontal line test
  • Algebra errors: Be careful when solving for y in complex functions

💡 Tips & Tricks

  • Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is NOT invertible
  • Domain & Range Swap: Domain of f = Range of f⁻¹, and Range of f = Domain of f⁻¹
  • Graphical Insight: The graph of f⁻¹ is the reflection of f's graph across the line y = x
  • Quick Check: Strictly increasing/decreasing functions are always invertible

Practice Suggestions

Start with linear functions, then progress to rational functions (like f(x) = (2x+1)/(x-3)) and radical functions. Practice both algebraic verification and graphical analysis. Try composing f(f⁻¹(x)) to verify your answers. Work with functions that require domain restrictions to become invertible.

Practice problems

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

1 f(x) = (x - 4)² + 2; restrict domain to make invertible

Hint: Consider the vertex of the parabola and which side creates a one-to-one function

Show the answer

Answer: x ≥ 4

  1. f(x) = (x - 4)² + 2 is a parabola opening upward with vertex at (4, 2)
  2. The function fails the horizontal line test because for most y-values, there are two x-values
  3. To make it invertible, restrict to either x ≥ 4 or x ≤ 4
  4. The standard restriction is x ≥ 4 (the right side of the vertex)
  5. With domain x ≥ 4, the function passes the horizontal line test and becomes invertible The restricted domain is x ≥ 4.

2 f(x) = (x - 6)² + 2; restrict domain to make invertible

Hint: Consider where the function changes direction and choose the side where each output corresponds to exactly one input

Show the answer

Answer: x ≥ 6

  1. The function f(x) = (x - 6)² + 2 is a parabola opening upward with vertex at (6, 2)
  2. Since it's a parabola, it fails the horizontal line test over its entire domain
  3. To make it invertible, we restrict to either x ≥ 6 (right side of vertex) or x ≤ 6 (left side of vertex)
  4. The standard convention is to restrict to x ≥ 6, which gives the increasing portion of the parabola
  5. With domain x ≥ 6, each y-value corresponds to exactly one x-value, making the function invertible
  6. The restricted domain is x ≥ 6

3 f(x) = (x - 6)² + 1. Restrict the domain to make it invertible.

Hint: Consider where the function changes from decreasing to increasing, and choose the side where it passes the horizontal line test.

Show the answer

Answer: x ≥ 6

  1. The function f(x) = (x - 6)² + 1 is a parabola opening upward with vertex at (6, 1).
  2. This parabola fails the horizontal line test over its entire domain because it is symmetric about x = 6.
  3. To make it invertible, we restrict the domain to either x ≥ 6 (right side of vertex) or x ≤ 6 (left side of vertex).
  4. The standard convention is to restrict to x ≥ 6, making the function one-to-one and invertible.
  5. Therefore, the domain restriction is x ≥ 6.

4 f(x) = (x - 9)² + 4. Find the domain restriction that makes f(x) invertible.

Hint: Consider where the function changes from decreasing to increasing, and think about which part would pass the horizontal line test.

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Answer: x ≥ 9

  1. The function f(x) = (x - 9)² + 4 is a parabola opening upward with vertex at (9, 4).
  2. Since it's a parabola, it fails the horizontal line test over its entire domain.
  3. To make it invertible, we need to restrict to either x ≥ 9 (right side of vertex) or x ≤ 9 (left side of vertex).
  4. The standard domain restriction for quadratic functions is typically the right side of the vertex, which gives x ≥ 9.
  5. With this restriction, the function becomes one-to-one and therefore invertible. The domain restriction is x ≥ 9.

5 f(x) = (x - 8)² is not invertible. Restrict the domain to make it invertible.

Hint: Consider which part of the parabola would pass the horizontal line test. Think about the vertex location and which side creates a one-to-one function.

Show the answer

Answer: x ≥ 8

  1. The function f(x) = (x - 8)² is a parabola opening upward with vertex at (8, 0).
  2. Since it's a parabola, it fails the horizontal line test - horizontal lines intersect it in two points.
  3. To make it invertible, we need to restrict to either x ≥ 8 (right side of vertex) or x ≤ 8 (left side of vertex).
  4. The standard convention is to restrict to x ≥ 8, which gives the increasing portion of the parabola.
  5. With domain x ≥ 8, the function passes the horizontal line test and becomes invertible. The restricted domain is x ≥ 8.

6 f(x) = (x - 2)² + 7; find the domain restriction x ≥ a that makes f invertible

Hint: Consider where the vertex of the parabola is located and which side of the vertex will make the function one-to-one

Show the answer

Answer: 2

  1. The function f(x) = (x - 2)² + 7 is a parabola opening upward
  2. The vertex is at x = 2 (from the term (x - 2)²)
  3. To make the function invertible, we need to restrict the domain to either x ≥ 2 or x ≤ 2
  4. The problem asks for x ≥ a, so we choose the right side of the vertex
  5. Therefore, a = 2, and the domain restriction is x ≥ 2
  6. With this restriction, f passes the horizontal line test and is invertible

The answer is 2.

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