Inverse Functions: Undoing What a Function Does

You encrypt a message. Your friend decrypts it.

You compress a file. You decompress it.

You convert dollars to euros. You convert euros back to dollars.

These are inverse operations.

An inverse function reverses what the original function does. If f takes 3 to 9, then f⁻¹ takes 9 back to 3.

The Unlock: Functions Can Run Backward

A function is a process: input goes in, output comes out.

An inverse function reverses the process.

If f(x) = y, then f⁻¹(y) = x.

The function maps x to y. The inverse maps y back to x.

Not every function has an inverse. But when it does, the inverse undoes the original.

The Definition: f⁻¹(f(x)) = x

If f and f⁻¹ are inverses, then:

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

f(f⁻¹(y)) = y for all y in the range of f.

Composing a function with its inverse gives the identity function.

The function and its inverse cancel each other out.

Example: f(x) = 2x. The inverse is f⁻¹(x) = x/2.

f(f⁻¹(x)) = f(x/2) = 2(x/2) = x. ✓

f⁻¹(f(x)) = f⁻¹(2x) = (2x)/2 = x. ✓

They undo each other.

How to Find an Inverse

Step 1: Write y = f(x).

Step 2: Solve for x in terms of y.

Step 3: Swap x and y. The result is f⁻¹(x).

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

Step 1: y = 2x + 3.

Step 2: Solve for x. y - 3 = 2x, so x = (y - 3)/2.

Step 3: Swap. f⁻¹(x) = (x - 3)/2.

Check: f(f⁻¹(x)) = f((x - 3)/2) = 2((x - 3)/2) + 3 = (x - 3) + 3 = x. ✓

Why Not Every Function Has an Inverse

For f⁻¹ to exist, every output of f must come from exactly one input.

If two different inputs give the same output, which one does the inverse return?

There's ambiguity. So the inverse doesn't exist.

Example: f(x) = x². Both f(2) = 4 and f(-2) = 4.

If f⁻¹(4) exists, should it equal 2 or -2? There's no unique answer.

So f(x) = x² does not have an inverse (over all real numbers).

The One-to-One Requirement

A function has an inverse if and only if it is one-to-one (injective).

One-to-one: Different inputs produce different outputs. If f(a) = f(b), then a = b.

Equivalently: No horizontal line crosses the graph more than once. (Horizontal line test.)

Example: f(x) = x³ is one-to-one. If a³ = b³, then a = b. So f has an inverse.

Example: f(x) = x² is not one-to-one (over all reals). 2² = 4 and (-2)² = 4. Two inputs give the same output.

Restricting the Domain

If a function is not one-to-one, you can sometimes restrict the domain to make it one-to-one.

Example: f(x) = x² is not one-to-one over all real numbers.

But restrict to x ≥ 0. Now f is one-to-one. Each output comes from exactly one non-negative input.

The inverse is f⁻¹(x) = √x, defined for x ≥ 0.

This is why the square root function is defined only for non-negative inputs: it's the inverse of x² restricted to x ≥ 0.

Graphical Property: Symmetry Across y = x

The graphs of f and f⁻¹ are reflections of each other across the line y = x.

Why? Because the inverse swaps x and y.

If (a, b) is on the graph of f, then f(a) = b, so f⁻¹(b) = a. That means (b, a) is on the graph of f⁻¹.

The points (a, b) and (b, a) are reflections across y = x.

Example: f(x) = 2x has points (1, 2), (2, 4), (3, 6).

f⁻¹(x) = x/2 has points (2, 1), (4, 2), (6, 3).

Each point on f⁻¹ is the reflection of a point on f.

Domain and Range Swap

If f has domain D and range R, then f⁻¹ has domain R and range D.

The inverse swaps inputs and outputs. So it swaps domain and range.

Example: f(x) = x² (restricted to x ≥ 0) has domain [0, ∞) and range [0, ∞).

f⁻¹(x) = √x has domain [0, ∞) and range [0, ∞).

Example: f(x) = e^x has domain (-∞, ∞) and range (0, ∞).

f⁻¹(x) = ln(x) has domain (0, ∞) and range (-∞, ∞).

The domain of the inverse is the range of the original.

Inverses of Linear Functions

A linear function f(x) = mx + b (with m ≠ 0) is one-to-one. It has an inverse.

To find it:

y = mx + b.

Solve for x: x = (y - b)/m.

Swap: f⁻¹(x) = (x - b)/m = (1/m)x - b/m.

The inverse is also linear.

The slope of f is m. The slope of f⁻¹ is 1/m.

Steep functions have flat inverses. Flat functions have steep inverses.

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

f⁻¹(x) = (x - 2)/3 = (1/3)x - 2/3.

Slope of f: 3. Slope of f⁻¹: 1/3.

Inverses of Power Functions

f(x) = x^n (for odd n) is one-to-one. The inverse is f⁻¹(x) = x^(1/n).

Example: f(x) = x³. The inverse is f⁻¹(x) = ∛x.

For even n, you need to restrict the domain.

f(x) = x² (x ≥ 0) has inverse f⁻¹(x) = √x.

f(x) = x² (x ≤ 0) has inverse f⁻¹(x) = -√x.

Inverses of Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other.

f(x) = e^x has inverse f⁻¹(x) = ln(x).

f(x) = 10^x has inverse f⁻¹(x) = log₁₀(x).

f(x) = b^x has inverse f⁻¹(x) = log_b(x).

This is why logarithms exist: they're the inverses of exponentials.

The equation e^x = 5 is solved by taking the logarithm: x = ln(5).

The logarithm undoes the exponential.

Inverses of Trigonometric Functions

Trig functions are periodic. They repeat. So they're not one-to-one.

But you can restrict the domain to make them one-to-one.

sin(x): Restrict to [-π/2, π/2]. Inverse: arcsin(x) or sin⁻¹(x).

cos(x): Restrict to [0, π]. Inverse: arccos(x) or cos⁻¹(x).

tan(x): Restrict to (-π/2, π/2). Inverse: arctan(x) or tan⁻¹(x).

The inverse trig functions are defined on these restricted domains.

Example: sin(π/6) = 1/2. So arcsin(1/2) = π/6.

But sin(5π/6) = 1/2 also. Why doesn't arcsin(1/2) = 5π/6?

Because arcsin is defined on the restricted domain [-π/2, π/2]. And 5π/6 is outside that range.

The restriction makes the inverse unique.

Notation: f⁻¹(x) vs. 1/f(x)

Warning: f⁻¹(x) does not mean 1/f(x).

f⁻¹(x) is the inverse function. 1/f(x) is the reciprocal.

They're different.

Example: f(x) = 2x.

f⁻¹(x) = x/2. (The inverse.)

1/f(x) = 1/(2x). (The reciprocal.)

f⁻¹(x) ≠ 1/f(x).

The notation is confusing, but standard. Context determines meaning.

Composition Property

If f and f⁻¹ are inverses:

(f ∘ f⁻¹)(x) = f(f⁻¹(x)) = x.

(f⁻¹ ∘ f)(x) = f⁻¹(f(x)) = x.

The compositions are the identity function.

This is the defining property of inverses.

Inverses and Solving Equations

Finding an inverse is equivalent to solving an equation.

To find f⁻¹(x), you solve y = f(x) for x in terms of y.

Example: Solve e^x = 5 for x.

This is asking: What is the inverse of e^x evaluated at 5?

x = ln(5).

Inverse functions are the tool for solving equations.

When Inverses Don't Exist: Horizontal Line Test

To check if f has an inverse, use the horizontal line test.

If any horizontal line crosses the graph of f more than once, f is not one-to-one. It doesn't have an inverse.

Example: f(x) = x² fails the horizontal line test. The line y = 4 crosses at x = 2 and x = -2.

So f(x) = x² (over all reals) has no inverse.

Piecewise Functions and Inverses

A piecewise function can have an inverse if it's one-to-one.

Example: f(x) = 2x if x < 0 f(x) = x + 1 if x ≥ 0

Check: Is this one-to-one?

For x < 0, f(x) = 2x gives outputs in (-∞, 0).

For x ≥ 0, f(x) = x + 1 gives outputs in [1, ∞).

The ranges don't overlap. No output comes from two inputs. So f is one-to-one.

To find the inverse, invert each piece.

For y < 0, y = 2x, so x = y/2. Thus f⁻¹(y) = y/2.

For y ≥ 1, y = x + 1, so x = y - 1. Thus f⁻¹(y) = y - 1.

f⁻¹(y) = y/2 if y < 0 f⁻¹(y) = y - 1 if y ≥ 1

(Swap y with x for standard notation.)

Inverses of Compositions

If f and g are invertible, then (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹.

The inverse of a composition is the composition of the inverses in reverse order.

Why? Because (f ∘ g) applies g first, then f. To undo it, you undo f first, then undo g.

Example: f(x) = 2x, g(x) = x + 3.

(f ∘ g)(x) = f(x + 3) = 2(x + 3) = 2x + 6.

Inverse: (2x + 6 - 6)/2 = x. So (f ∘ g)⁻¹(x) = (x - 6)/2.

Using the formula:

f⁻¹(x) = x/2, g⁻¹(x) = x - 3.

(g⁻¹ ∘ f⁻¹)(x) = g⁻¹(x/2) = x/2 - 3 = (x - 6)/2. ✓

They match.

Why Inverses Matter: Reversibility

Inverses encode reversibility.

If a process is reversible, it has an inverse function.

Encryption: encrypt with f, decrypt with f⁻¹.

Unit conversion: convert meters to feet with f, feet to meters with f⁻¹.

Coordinate transformations: rotate coordinates with f, rotate back with f⁻¹.

Inverses are how you undo operations.

Inverses and Bijectivity

A function is bijective (one-to-one and onto) if every element of the codomain is the image of exactly one element of the domain.

Bijective functions have inverses.

One-to-one ensures uniqueness. Onto ensures existence.

In practice, for functions from ℝ to ℝ, you need:

• One-to-one: different inputs give different outputs.
• Onto: every possible output is achieved.

Example: f(x) = x³ is bijective (from ℝ to ℝ). Every real number is the cube of some real number. And cubing is one-to-one.

So f⁻¹(x) = ∛x exists.

Practical Example: Temperature Conversion

Convert Celsius to Fahrenheit: F = (9/5)C + 32.

To convert Fahrenheit back to Celsius, you need the inverse.

Solve for C:

F = (9/5)C + 32

F - 32 = (9/5)C

C = (5/9)(F - 32).

The inverse is C = (5/9)(F - 32).

This is the formula for converting Fahrenheit to Celsius.

Inverse functions are everywhere in applied math.

Self-Inverse Functions

Some functions are their own inverse.

f(f(x)) = x.

Example: f(x) = 1/x (for x ≠ 0).

f(f(x)) = f(1/x) = 1/(1/x) = x.

So f is its own inverse. f⁻¹ = f.

Example: f(x) = -x.

f(f(x)) = f(-x) = -(-x) = x.

Negation is its own inverse.

These functions are called involutions.

Derivative of the Inverse

In calculus, there's a formula for the derivative of an inverse function.

If f is differentiable and invertible, then:

(f⁻¹)'(y) = 1 / f'(f⁻¹(y)).

The slope of the inverse is the reciprocal of the slope of the original.

Example: f(x) = x³. f'(x) = 3x².

f⁻¹(x) = ∛x. (f⁻¹)'(x) = 1/(3(∛x)²) = 1/(3x^(2/3)).

Check: The derivative of x^(1/3) is (1/3)x^(-2/3) = 1/(3x^(2/3)). ✓

Implicit vs. Explicit Inverses

Sometimes you can't solve y = f(x) for x explicitly.

Example: y = x + e^x.

There's no closed-form solution for x in terms of y.

But the function is one-to-one (it's strictly increasing). So the inverse exists.

We say the inverse is implicit. It exists, but we can't write a formula for it.

In practice, you compute values numerically.

Numerical Inverses

For functions without explicit inverses, you use numerical methods.

Example: Solve x + e^x = 2 for x.

This is asking for f⁻¹(2), where f(x) = x + e^x.

You use Newton's method or a similar algorithm to approximate x ≈ 0.4428.

Inverse functions don't always have nice formulas, but they still exist conceptually.

Why Inverses Are Conceptually Important

Inverses are about symmetry and reversibility.

If you can reverse a process, you understand it more deeply.

In abstract algebra, groups are defined by having inverses. Every element has an inverse under the operation.

In linear algebra, invertible matrices are central. They represent reversible transformations.

In calculus, the fundamental theorem relates integrals and derivatives—they're inverses in a sense.

Inverses are a unifying concept across mathematics.

Common Mistakes

Mistake 1: Confusing f⁻¹(x) with 1/f(x).

f⁻¹(x) is the inverse. 1/f(x) is the reciprocal. Different concepts.

Mistake 2: Forgetting to restrict the domain.

f(x) = x² has no inverse over all reals. You must restrict to x ≥ 0 or x ≤ 0.

Mistake 3: Swapping x and y too early.

First solve for x in terms of y. Then swap. Don't swap first.

Mistake 4: Assuming every function has an inverse.

Only one-to-one functions have inverses. Use the horizontal line test.

Mistake 5: Forgetting that domain and range swap.

The domain of f⁻¹ is the range of f. The range of f⁻¹ is the domain of f.

The Payoff: Undoing Processes

When you understand inverse functions, you see operations and their undoings as a symmetric pair.

You stop thinking: "How do I solve this equation?"

You start thinking: "What function do I need to invert?"

That shift—from solving as a procedure to inverting as a concept—is powerful.

It's how you solve exponential equations (with logarithms), how you solve trig equations (with inverse trig), how you reverse coordinate transformations.

Inverses are about seeing structure, not memorizing formulas.

Part 5 of the Precalculus series.

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