Core Idea (Inverse Function):
\( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)
General Steps to Find an Inverse:
- Write \( y = f(x) \)
- Swap variables: \( x = f(y) \)
- Solve for \( y \)
- 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
- Select a function type from the dropdown (e.g., Linear, Quadratic, Exponential).
- Enter the parameters (like \( a, b, c \)). For “Custom,” type an expression such as
2*x+3
,e^x
,sqrt(x)
, orlog(x)
. - 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.
- 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.
- 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