Inverse Functions: Find Algebraically

Grade 12 · algebra · 88 practice problems · read aloud

🔊 Listen to this explanation

Inverse Functions: Unlocking the Reverse Process

An inverse function, denoted as f⁻¹(x), essentially reverses the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a. They are crucial for solving equations, understanding logarithmic and exponential relationships, and modeling real-world scenarios where you need to work backwards from an output to an input.

🔍 Step-by-Step Guide to Finding an Inverse

  1. Replace f(x): Write the function as y = f(x).
  2. Swap Variables: Interchange x and y.
  3. Solve for y: Algebraically isolate the new y.
  4. Apply Notation: Replace y with f⁻¹(x).
  5. Verify: Check that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

📚 Worked Examples

Example 1: Linear Function
Find the inverse of f(x) = 2x - 6.

  1. y = 2x - 6
  2. x = 2y - 6
  3. x + 6 = 2y → y = (x + 6)/2
  4. f⁻¹(x) = (x + 6)/2

Example 2: Rational Function
Find the inverse of f(x) = (3x + 1)/(x - 4), x ≠ 4.

  1. y = (3x + 1)/(x - 4)
  2. x = (3y + 1)/(y - 4)
  3. x(y - 4) = 3y + 1 → xy - 4x = 3y + 1 → xy - 3y = 4x + 1 → y(x - 3) = 4x + 1
  4. y = (4x + 1)/(x - 3)
  5. f⁻¹(x) = (4x + 1)/(x - 3), x ≠ 3

⚠️ Common Mistakes to Avoid

  • Incorrect Notation: f⁻¹(x) does NOT mean 1/f(x). It is the inverse function, not the reciprocal.
  • Forgetting to Swap Domain & Range: The domain of f becomes the range of f⁻¹, and vice-versa.
  • Not Checking for One-to-One: A function must be one-to-one (pass the horizontal line test) to have an inverse.
  • Algebra Errors: Be meticulous when solving for y, especially with rational functions.

💡 Tips & Tricks

  • Graphical Insight: The graph of f⁻¹(x) is the reflection of f(x) over the line y = x.
  • Composition Check: Always verify your inverse using f(f⁻¹(x)) = x.
  • Domain First: Before starting, identify the domain restrictions of the original function.

🎯 Practice Suggestions

Start with simple linear functions, then progress to rational, square root, and quadratic (with restricted domains). Create a table of values for a function and its inverse to see the input-output reversal. Use graphing software to visualize the reflection over y=x. Practice verifying your answers with the composition property.

Practice problems

6 of the 88, worked through step by step — try them before opening the answer.

1 f(x) = 3x - 7, find f⁻¹(x) = ?

Hint: To find an inverse function algebraically, replace f(x) with y, then swap x and y variables, and solve for y. For example, if g(x) = 2x + 5, you would write y = 2x + 5, then x = 2y + 5, then solve for y.

Show the answer

Answer: (x + 7)/3

  1. Replace f(x) with y This makes the equation easier to work with. So we have: y = 3x - 7
  2. Swap x and y For an inverse function, the roles of inputs and outputs are reversed. After swapping we get: x = 3y - 7
  3. Solve for y We need to isolate y on one side of the equation. First, add 7 to both sides: x + 7 = 3y - 7 + 7 x + 7 = 3y Next, divide both sides by 3: (x + 7)/3 = 3y/3 (x + 7)/3 = y
  4. Write the inverse function Since we now have y = (x + 7)/3, and y represents the inverse function, we can write: f⁻¹(x) = (x + 7)/3 Therefore, the inverse function is f⁻¹(x) = (x + 7)/3

To find the inverse function f⁻¹(x) for f(x) = 3x - 7, follow these steps:

2 f(x) = 2 + √(3x - 1), find f⁻¹(x) = ?

Hint: To find an inverse function, start by replacing f(x) with y, then swap x and y, and solve for y. Remember to consider the domain restrictions of the original function.

Show the answer

Answer: (x² - 4x + 5)/3

  1. Replace f(x) with y: y = 2 + √(3x - 1)
  2. Swap x and y: x = 2 + √(3y - 1)
  3. Isolate the square root: x - 2 = √(3y - 1)
  4. Square both sides: (x - 2)² = 3y - 1
  5. Expand the left side: x² - 4x + 4 = 3y - 1
  6. Solve for y: 3y = x² - 4x + 4 + 1
  7. Simplify: 3y = x² - 4x + 5
  8. Divide by 3: y = (x² - 4x + 5)/3
  9. The inverse function is f⁻¹(x) = (x² - 4x + 5)/3 Note: The domain of f⁻¹(x) is x ≥ 2 since the original function f(x) has range [2, ∞).

3 f(x) = (2x - 3)/(x + 1), find f⁻¹(x) = ?

Hint: To find an inverse function algebraically, replace f(x) with y, swap x and y, then solve for y. For a rational function like g(t) = (at + b)/(ct + d), you'll need to isolate the variable term.

Show the answer

Answer: f⁻¹(x) = (x + 3)/(2 - x)

  1. Replace f(x) with y** y = (2x - 3)/(x + 1) --- **
  2. Swap x and y** To find the inverse, we swap x and y: x = (2y - 3)/(y + 1) --- **
  3. Solve for y** Multiply both sides by (y + 1): x(y + 1) = 2y - 3 Expand left side: xy + x = 2y - 3 --- **
  4. Get all terms with y on one side** xy - 2y = -3 - x Factor y on the left: y(x - 2) = -3 - x --- **
  5. Solve for y** y = (-3 - x)/(x - 2) --- **
  6. Simplify** Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) --- **
  7. Write final inverse function** f⁻¹(x) = (x + 3)/(2 - x) --- **Final answer:** f⁻¹(x) = (x + 3)/(2 - x)

Let's find the inverse function step by step. We are given: f(x) = (2x - 3)/(x + 1) --- **

4 f(x) = (3x - 2)/(x + 4), find f⁻¹(x) = ?

Hint: To find an inverse function algebraically, replace f(x) with y, swap x and y, then solve for y. For a rational function like (ax + b)/(cx + d), you'll need to isolate y using algebraic manipulation.

Show the answer

Answer: f⁻¹(x) = (-4x - 2)/(x - 3)

  1. Replace f(x) with y** y = (3x - 2)/(x + 4) --- **
  2. Swap x and y** To find the inverse, we swap x and y: x = (3y - 2)/(y + 4) --- **
  3. Solve for y** Multiply both sides by (y + 4): x(y + 4) = 3y - 2 Expand left side: xy + 4x = 3y - 2 --- **
  4. Get all terms with y on one side** xy - 3y = -4x - 2 Factor y on the left: y(x - 3) = -4x - 2 --- **
  5. Solve for y** y = (-4x - 2)/(x - 3) --- **
  6. Write the inverse function** f⁻¹(x) = (-4x - 2)/(x - 3) --- **Final answer:** f⁻¹(x) = (-4x - 2)/(x - 3)

Let's find the inverse function step by step. We are given: f(x) = (3x - 2)/(x + 4) --- **

5 f(x) = (2x + 3)/(x - 1), find f⁻¹(x) = ?

Hint: To find an inverse function algebraically, replace f(x) with y, swap x and y, then solve for y. Remember to verify that your result is indeed the inverse by checking function composition.

Show the answer

Answer: (x + 3)/(x - 2)

  1. Replace f(x) with y** y = (2x + 3)/(x - 1) --- **
  2. Swap x and y** To find the inverse, we swap x and y: x = (2y + 3)/(y - 1) --- **
  3. Solve for y** Multiply both sides by (y - 1): x(y - 1) = 2y + 3 Distribute x: xy - x = 2y + 3 --- **
  4. Get all y terms on one side** xy - 2y = x + 3 Factor y on the left: y(x - 2) = x + 3 --- **
  5. Solve for y** y = (x + 3)/(x - 2) --- **
  6. Write the inverse function** f⁻¹(x) = (x + 3)/(x - 2) --- **Final answer:** f⁻¹(x) = (x + 3)/(x - 2)

Let's find the inverse function step by step. We start with: f(x) = (2x + 3)/(x - 1) --- **

6 f(x) = (2x - 5)/(x + 3), find f⁻¹(x) = ?

Hint: To find the inverse of a rational function, swap the variables and then solve for the new output variable. Consider clearing fractions by multiplying both sides by the denominator.

Show the answer

Answer: f⁻¹(x) = (-3x - 5)/(x - 2)

  1. Replace f(x) with y y = (2x - 5)/(x + 3)
  2. Swap x and y This is the key step for finding an inverse function - we exchange the roles of x and y: x = (2y - 5)/(y + 3)
  3. Solve for y We need to isolate y on one side of the equation. First, multiply both sides by (y + 3) to eliminate the denominator: x(y + 3) = 2y - 5
  4. Distribute x on the left side: xy + 3x = 2y - 5
  5. Get all terms with y on one side and other terms on the other side: xy - 2y = -3x - 5
  6. Factor y out of the left side: y(x - 2) = -3x - 5
  7. Divide both sides by (x - 2) to solve for y: y = (-3x - 5)/(x - 2)
  8. Write the final answer Since y now represents f⁻¹(x), we have: f⁻¹(x) = (-3x - 5)/(x - 2) This is the inverse function of f(x).

To find the inverse function f⁻¹(x) for f(x) = (2x - 5)/(x + 3), follow these steps:

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