3.8Radical Functions and Inverse Relationships
Find inverses of polynomial functions, graph radical functions, restrict domains to ensure invertibility, and solve radical equations.
Radical functions are the inverses of power functions. Mastering them — including domain restrictions and simplification — is essential for solving equations involving roots and for understanding inverse relationships in calculus.
Essential Question
How do you find the inverse of a function, and why must some functions have their domain restricted before an inverse can exist?
Lesson Overview
In this lesson you will learn how to find the inverse of a function by swapping x and y and solving for y. You will explore which functions have inverses (one-to-one functions that pass the horizontal line test), how to restrict domains to create invertible functions, and how radical functions arise naturally as inverses of power functions. You will also solve radical equations and verify inverses using composition.
Finding an Inverse
- 1Replace f(x) with y.
- 2Swap x and y.
- 3Solve the new equation for y.
- 4Replace y with f⁻¹(x).
- 5State the domain of f⁻¹.
Key Facts
- •A function has an inverse iff it is one-to-one (passes the horizontal line test).
- •f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the respective domains.
- •The graph of f⁻¹ is the reflection of f over the line y = x.
- •Even-index radicals (√x, x^(1/4)) require domain x ≥ 0.
- •Odd-index radicals (∛x, x^(1/5)) have domain all real numbers.
f(x) = x² (x ≥ 0) and f⁻¹(x) = √x reflected over y = x
Radical Function Family: y = √x, y = ∛x, y = x^(1/4)
Solving √(2x+3) = 5 — Step by Step
Worked Examples
Find the inverse of f(x) = 3x − 7.
Step 1: Write y = 3x − 7.
Step 2: Swap x and y: x = 3y − 7.
Step 3: Solve for y: x + 7 = 3y → y = (x + 7) / 3.
Step 4: Write f⁻¹(x) = (x + 7) / 3.
Verify: f(f⁻¹(x)) = 3·((x+7)/3) − 7 = x + 7 − 7 = x ✓
Find the inverse of f(x) = x² with domain restricted to x ≥ 0.
Step 1: Write y = x², x ≥ 0.
Step 2: Swap x and y: x = y², y ≥ 0.
Step 3: Solve for y: y = √x (take positive root since y ≥ 0).
Step 4: Write f⁻¹(x) = √x, domain x ≥ 0.
Verify: f(f⁻¹(x)) = (√x)² = x for x ≥ 0 ✓
Solve the radical equation √(2x + 3) = 5.
Step 1: Isolate the radical — it is already isolated.
Step 2: Square both sides: (√(2x+3))² = 5² → 2x + 3 = 25.
Step 3: Solve: 2x = 22 → x = 11.
Step 4: Check for extraneous solutions: √(2·11+3) = √25 = 5 ✓
Determine the domain and range of f(x) = √(x − 4) + 2.
Domain: the radicand must be ≥ 0: x − 4 ≥ 0 → x ≥ 4.
Range: √(x−4) ≥ 0, so f(x) = √(x−4) + 2 ≥ 2.
Domain: [4, ∞). Range: [2, ∞).
Find the inverse of f(x) = (x − 1)³ + 2.
Step 1: Write y = (x − 1)³ + 2.
Step 2: Swap x and y: x = (y − 1)³ + 2.
Step 3: Isolate the cube: x − 2 = (y − 1)³.
Step 4: Take the cube root: ∛(x − 2) = y − 1.
Step 5: Solve: y = ∛(x − 2) + 1.
No domain restriction needed — f is one-to-one (odd power).
Guided Practice
Find the inverse of f(x) = 5x + 2.
Hint: Swap x and y, then solve for y. The inverse will be a linear function.
Find the inverse of f(x) = x³ − 4.
Hint: After swapping, isolate the cube term, then take the cube root of both sides.
Solve: √(3x − 6) = 3.
Hint: Square both sides to eliminate the radical, then solve the resulting linear equation. Don't forget to check for extraneous solutions.
State the domain of f(x) = √(5 − 2x).
Hint: Set the radicand 5 − 2x ≥ 0 and solve the inequality for x.
Verify that f(x) = 2x + 6 and g(x) = (x − 6)/2 are inverses of each other.
Hint: Compute f(g(x)) and g(f(x)). Both must equal x.
Key Vocabulary
Inverse Function
A function f⁻¹ that reverses the action of f. If f(a) = b, then f⁻¹(b) = a.
Example: f(x) = 2x + 1 → f⁻¹(x) = (x − 1)/2
One-to-One Function
A function where each output value corresponds to exactly one input. Passes the horizontal line test.
Example: f(x) = x³ is one-to-one; f(x) = x² is not (without restriction).
Horizontal Line Test
A graphical test: if every horizontal line intersects the graph at most once, the function is one-to-one and has an inverse.
Example: y = √x passes; y = x² (unrestricted) fails.
Radical Function
A function containing a variable under a radical sign (root). The index n determines domain and range.
Example: f(x) = √x (n=2), f(x) = ∛x (n=3), f(x) = x^(1/4) (n=4)
Domain Restriction
Limiting the domain of a function so that it becomes one-to-one, allowing an inverse to exist.
Example: f(x) = x² restricted to x ≥ 0 has inverse f⁻¹(x) = √x.
Extraneous Solution
A solution obtained algebraically that does not satisfy the original equation. Always check solutions in radical equations.
Example: Squaring both sides of √x = −3 gives x = 9, but √9 = 3 ≠ −3 (extraneous).
Check Your Understanding
Interactive Practice — 5 Questions
What is the inverse of f(x) = 4x − 8?
Which function is the inverse of f(x) = x² when the domain is restricted to x ≥ 0?
What is the solution to √(x + 5) = 4?
What is the domain of f(x) = √(2x − 10)?
Which statement about the cube root function f(x) = ∛x is TRUE?
Independent Practice
Independent Practice
Find the inverse of f(x) = 7x − 3.
Find the inverse of f(x) = x³ + 5.
Solve: √(4x − 8) = 6.
State the domain and range of f(x) = √(x + 9) − 3.
Verify that f(x) = x³ − 1 and g(x) = ∛(x + 1) are inverses by computing f(g(x)) and g(f(x)).
Common Mistakes
Writing f⁻¹(x) = 1/f(x). The notation f⁻¹ means the inverse function, NOT the reciprocal.
f⁻¹(x) is found by swapping x and y and solving. For example, if f(x) = 2x, then f⁻¹(x) = x/2, not 1/(2x).
Forgetting to check for extraneous solutions after squaring both sides of a radical equation.
Always substitute your answer back into the original equation. Squaring can introduce false solutions.
Assuming every function has an inverse without checking the horizontal line test.
Only one-to-one functions have inverses. Restrict the domain of f(x) = x² to x ≥ 0 before finding its inverse.
Applying even-root domain rules to odd roots: writing "∛x requires x ≥ 0."
Odd-index radicals (∛x, x^(1/5), etc.) are defined for all real numbers, including negatives.
Math Tips
The graph of f⁻¹ is always the reflection of f over the line y = x. Use this to sketch inverses quickly.
To verify two functions are inverses, compute both f(g(x)) and g(f(x)). Both must simplify to x.
For domain of √(expression): set expression ≥ 0 and solve. For ∛(expression): domain is always all reals.
When solving radical equations, isolate the radical first before raising both sides to a power.
nth root functions: even n → domain x ≥ 0, range y ≥ 0. Odd n → domain all reals, range all reals.