Finding Function Inverses- g² Calculation Methods

What Function Inverses Actually Are

A function inverse is what you get when you swap the input and output of a function. If f(x) takes 3 to 9, then f⁻¹(9) gives you 3 back. That's the whole idea. Nothing mystical about it.

You write it as f⁻¹(x), but don't confuse this with reciprocals. This symbol means "inverse function," not "one over f(x)." Students mix these up constantly.

The Core Rule: Swap and Solve

Every method for finding inverses follows the same logic:

1. Replace f(x) with y
2. Swap x and y
3. Solve for the new y
4. Replace y with f⁻¹(x)

That's it. Every problem, every function, every time. If you're struggling with inverses, you're probably skipping step 2 or solving for the wrong variable.

Simple Example: Linear Functions

Take f(x) = 3x + 7

Step 1: y = 3x + 7

Step 2: x = 3y + 7 (swap x and y)

Step 3: x - 7 = 3y

Step 4: y = (x - 7)/3

Result: f⁻¹(x) = (x - 7)/3

Linear inverses are straightforward because you're just isolating the variable on the other side.

Squared Functions: Where It Gets Messy

When you see g² or g(x) = x², the inverse isn't a function anymore—at least not one you can use without restrictions.

g(x) = x² gives you the same output for +3 and -3. The inverse relation would map 9 back to both 3 and -3. That's not a function by definition.

To make g² invertible, you must restrict the domain:

• g(x) = x², x ≥ 0 → g⁻¹(x) = √x
• g(x) = x², x ≤ 0 → g⁻¹(x) = -√x

Pick one. You can't have both and still call it a function.

Methods for g² Calculation

When working with squared functions and their inverses, you have two practical approaches:

Method 1: Algebraic Restriction

State the domain restriction upfront, then apply the swap-and-solve method.

Example: g(x) = (x - 2)² + 5, with x ≥ 2

1. y = (x - 2)² + 5
2. x = (y - 2)² + 5
3. x - 5 = (y - 2)²
4. √(x - 5) = y - 2
5. y = 2 + √(x - 5)

Result: g⁻¹(x) = 2 + √(x - 5)

Method 2: Graphical Reflection

Graph y = f(x), then reflect across the line y = x. The reflected curve is your inverse relation. This works for any function and gives you a visual check.

Comparison: When to Use Each Method

Verification: The Composition Test

After you find an inverse, verify it. The definition requires:

• f(f⁻¹(x)) = x (for all x in the domain of f⁻¹)
• f⁻¹(f(x)) = x (for all x in the domain of f)

If either composition gives you anything other than x, your inverse is wrong. No exceptions.

Example check for f(x) = 3x + 7:

f(f⁻¹(x)) = 3((x-7)/3) + 7 = x - 7 + 7 = x ✓

f⁻¹(f(x)) = ((3x+7) - 7)/3 = 3x/3 = x ✓

Common Mistakes That Ruin Your Answer

• Forgetting domain restrictions on square roots and even-power functions
• Swapping incorrectly — some students write x = f(y) instead of swapping to y = f(x)
• Solving for x instead of y after the swap
• Not checking both compositions — one passing doesn't mean the inverse is correct

Getting Started: Your First Five Inverses

Practice this sequence to build the skill:

1. f(x) = 2x - 5 → f⁻¹(x) = (x + 5)/2
2. f(x) = (x + 3)/4 → f⁻¹(x) = 4x - 3
3. f(x) = x³ → f⁻¹(x) = ∛x
4. g(x) = √(x - 1), x ≥ 1 → g⁻¹(x) = x² + 1
5. h(x) = 5/(x + 2) → h⁻¹(x) = 5/x - 2

Work through each one without looking at the answers. Check with compositions. Repeat until you can do them without the steps written down.

When g² Appears in Composite Functions

If you're dealing with g²(g(x)), this means g(g(x))—the composition of g with itself, not g squared. The notation matters.

• g²(x) sometimes means g(g(x)) in function notation
• g(x)² means [g(x)]² — the output is squared

Context tells you which one applies. When in doubt, rewrite using composition notation instead of exponents.

What You Actually Need to Remember

Finding inverses is a three-step process: swap, solve, verify. The hard part isn't understanding the method—it's executing the algebra without mistakes.

For g² functions specifically, the domain restriction is non-negotiable. You cannot write a proper inverse for an even-power function without stating where it starts. Your textbook or instructor will expect this.

Master the linear cases first. Move to quadratics with restrictions. Then tackle everything else. You don't need to understand inverses intuitively—you just need to follow the steps correctly every time.