🔄 Inverse Function Calculator

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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

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