🔄 Inverse Function Calculator

Find the Inverse of a Function

The Inverse Function Calculator is designed to find the inverse of a given function, a common task in algebra and calculus. An inverse function, denoted as f⁻¹(x), essentially \"reverses\" the original function. If f(a) = b, then f⁻¹(b) = a. Our calculator not only provides the solution but can also show the steps involved in finding it.

Inverse Function Calculator

Find the inverse of a function algebraically and visualize both the original function and its inverse. This calculator supports common functions including linear, quadratic, exponential, logarithmic, and trigonometric functions.

Function Input

Linear function: f(x) = ax + b

Display Options

About Inverse Functions

An inverse function reverses the operation of the original function. If f(a) = b, then f⁻¹(b) = a. The inverse function "undoes" what the original function does.

Finding an Inverse Function
  1. Replace f(x) with y
  2. Switch x and y in the equation
  3. Solve for y
  4. Replace y with f⁻¹(x)
Conditions for Inverse Functions
  • One-to-one: A function must be one-to-one (injective) to have an inverse. Each output corresponds to exactly one input.
  • Horizontal Line Test: If any horizontal line intersects the graph more than once, the function is not one-to-one.
  • Domain and Range: The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.
Properties of Inverse Functions
  • 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
  • The graphs of f and f⁻¹ are reflections of each other across the line y = x
  • If (a, b) is on the graph of f, then (b, a) is on the graph of f⁻¹
Common Inverse Function Pairs
  • Linear: f(x) = ax + b → f⁻¹(x) = (x - b)/a
  • Exponential/Logarithmic: f(x) = aˣ → f⁻¹(x) = log_a(x)
  • Square/Square Root: f(x) = x² (x ≥ 0) → f⁻¹(x) = √x
  • Reciprocal: f(x) = 1/x → f⁻¹(x) = 1/x (self-inverse)
Applications
  • Temperature conversions (Celsius to Fahrenheit and vice versa)
  • Currency conversions
  • Encoding and decoding information
  • Solving equations
  • Physics and engineering calculations

Core Idea (Inverse Function):

\( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)

General Steps to Find an Inverse:

  1. Write \( y = f(x) \)
  2. Swap variables: \( x = f(y) \)
  3. Solve for \( y \)
  4. Rename \( y \) as \( f^{-1}(x) \)

Common Inverse Pairs (Quick Reference):

  • Linear: \( f(x)=ax+b \Rightarrow f^{-1}(x)=\frac{x-b}{a} \quad (a\neq 0) \)
  • Exponential / Logarithmic: \( f(x)=a\cdot b^{x} \Rightarrow f^{-1}(x)=\log_b\!\left(\frac{x}{a}\right) \quad (b>0, b\neq 1) \)
  • Logarithmic / Exponential: \( f(x)=a\log_b(x)+c \Rightarrow f^{-1}(x)= b^{\frac{x-c}{a}} \)
  • Radical: \( f(x)=a\sqrt{bx+c}+d \Rightarrow f^{-1}(x)=\frac{\left(\frac{x-d}{a}\right)^2-c}{b} \)
  • Rational (hyperbola): \( f(x)=\frac{a}{bx+c}+d \Rightarrow f^{-1}(x)=\frac{\frac{a}{x-d}-c}{b} \)

What the Inverse Function Calculator Does

This calculator finds the inverse of a function and shows how the two functions relate. It also plots both curves and the line \( y=x \) so you can see the reflection property clearly. You can work with linear, quadratic, cubic, exponential, logarithmic, radical, rational, or a custom function.

  • Algebra-first: See the inverse in standard notation.
  • Step-by-step: Follow clean, human-readable steps.
  • Graph view: Compare \( f(x) \), \( f^{-1}(x) \), and \( y=x \).
  • Verification table: Check that composition returns \( x \).
  • Properties: Review domain, range, asymptotes, and one-to-one status.

Why It’s Useful

Inverse functions let you reverse a calculation. This helps when you want to:

  • Convert units (e.g., Celsius ↔ Fahrenheit).
  • Solve equations by isolating the input from an output.
  • Interpret models in algebra, calculus, physics, and engineering.
  • Understand graphs using symmetry across \( y=x \).
  • Check injective behavior (one-to-one) with the horizontal line test.

How to Use the Calculator

  1. Select a function type from the dropdown (e.g., Linear, Quadratic, Exponential).
  2. Enter the parameters (like \( a, b, c \)). For “Custom,” type an expression such as 2*x+3, e^x, sqrt(x), or log(x).
  3. Adjust options:
    • Decimal Places to control numeric precision.
    • Graph Range (x) to set the plotting window.
    • Toggle Show calculation steps and Show graph as you like.
  4. Click “Find Inverse.” The app shows:
    • Original Function and Inverse Function in math form.
    • Verification values for \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \).
    • Graph with \( f \), \( f^{-1} \), and the line \( y=x \).
    • Properties such as domain, range, monotonic behavior, and any asymptote.
  5. Use “Reset” to restore defaults and start again.

Reading the Results

  • Algebraic form: The inverse shows how to recover the input from the output.
  • Graph: The curves for \( f \) and \( f^{-1} \) are reflections across \( y=x \).
  • Verification table: Values should match \( x \) within the shown precision.
  • Properties panel: Check domain and range swaps, and any restrictions.

Tips, Domains, and Restrictions

  • One-to-one is essential: A function must be injective on the chosen domain to have an inverse.
  • Quadratic note: You may need a domain restriction (e.g., \( x \ge h \) or \( x \le h \)) to make the inverse single-valued.
  • Exponential & logarithmic: Ensure \( b>0 \) and \( b\neq 1 \). Log inputs must be positive.
  • Radical functions: The expression under the square root must be nonnegative.
  • Rational forms: Watch for vertical and horizontal asymptotes; these affect domain and range.
  • Cubic forms: General inverses can be intricate; simplified forms invert cleanly.

Example Workflows

  • Linear: Set \( a=2, b=3 \). Result: \( f^{-1}(x)=\frac{x-3}{2} \). Use this to reverse a scaling-and-shift.
  • Exponential: With \( a=1, b=2 \), get \( f^{-1}(x)=\log_2(x) \). This turns outputs back into exponents.
  • Logarithmic: With \( a=1, b=10, c=0 \), get \( f^{-1}(x)=10^{x} \). This undoes base-10 logs.

FAQ

Q: What is an inverse function in simple terms?
A: It reverses the effect of the original function. If \( f \) sends \( a \) to \( b \), then \( f^{-1} \) sends \( b \) back to \( a \).

Q: Why does the graph of an inverse reflect across \( y=x \)?
A: Because every point \( (a,b) \) on \( f \) becomes \( (b,a) \) on \( f^{-1} \). Swapping coordinates mirrors across \( y=x \).

Q: What if my function fails the horizontal line test?
A: Restrict the domain to a region where the function is strictly monotonic (injective). Then compute the inverse on that region.

Q: What do “domain” and “range” do here?
A: They swap between \( f \) and \( f^{-1} \). The domain of \( f \) becomes the range of \( f^{-1} \), and vice versa.

Q: Can every function have an inverse?
A: No. A function must be one-to-one on the chosen domain. Otherwise multiple inputs could map to the same output.

Q: Why does the calculator show asymptotes for rational functions?
A: Asymptotes indicate values that the function approaches but never reaches. These values become exclusions in domains or ranges of \( f \) and \( f^{-1} \).

Q: What does the verification table prove?
A: It checks compositions numerically. If \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) return \( x \) (within rounding), the inverse is consistent.

Related Concepts Mentioned

  • Algebra, composition, bijection, injective, monotonic
  • Logarithm, exponential, radical, rational
  • Horizontal line test, reflection, asymptote, domain, range

More Information

How to Find an Inverse Function Manually:

The process involves a few straightforward algebraic steps:

  1. Replace f(x) with y. This makes the equation easier to work with.
  2. Swap the variables x and y. Every x becomes a y, and every y becomes an x.
  3. Solve the new equation for y. This is often the most challenging algebraic step.
  4. Replace the new y with f⁻¹(x). This is the final notation for the inverse function.

Our calculator automates this entire process for you.

Frequently Asked Questions

What is an inverse function?
An inverse function is a function that reverses another function. If the function "f" applied to an input "x" gives a result "y", then applying the inverse function "g" to "y" gives the result "x". So, g(f(x)) = x.
Does every function have an inverse?
No, a function must be "one-to-one" to have an inverse. This means that every input (x) corresponds to a unique output (y), and every output corresponds to a unique input. Functions that pass the Horizontal Line Test have an inverse.
What is the graph of an inverse function?
The graph of an inverse function f⁻¹(x) is a reflection of the graph of the original function f(x) across the line y = x.

About Us

We create powerful and intuitive math tools to help students learn and professionals work more efficiently. Our goal is to make complex mathematical concepts easier to understand by providing step-by-step solutions and clear explanations.

Contact Us