Functions & Transformations (Math 30-1)
Explore operations, compositions, and inverses of functions, plus graph transformations including stretches, reflections, and combinations.
Operations: \( (f\pm g)(x),\ (fg)(x),\ \frac{f}{g}(x) \)
Composition: \( (f\circ g)(x)=f(g(x)) \)
Inverse: \( f^{-1}(x) \)
Transform: \( g(x)=a\,f(b(x-h))+k \)
What You Should Be Able To Do
- Perform operations on functions and state domains appropriately.
- Build and evaluate compositions \( (f\circ g)(x) \) and determine domains.
- Verify whether a function has an inverse (horizontal line test) and interpret \( f^{-1} \).
- Describe and graph transformations \( g(x)=a\,f(b(x-h))+k \) including stretches, reflections, and translations.
- Communicate restrictions, intercepts, asymptotes, and key points clearly.
Progress autosaves to this device.
Video Library
Operations on Functions
Add, subtract, multiply, divide; domain restrictions.
youtube.com – “operations on functions algebra”Function Composition
Evaluate \( (f\circ g)(x) \), domains, input/output chains.
youtube.com – “function composition examples”Inverse Functions & HLT
One-to-one, restricting domain, checking inverses.
youtube.com – “finding inverse functions horizontal line test”Transformations \(a\,f(b(x-h))+k\)
Reflections, stretches/compressions, shifts, order of operations.
youtube.com – “function transformations a(b(x-h))+k”Interactive Tools
Transformation Explorer
Parent \(f(x)\) \(g(x)=a\,f(b(x-h))+k\)
Composition & Inverse Tester
\(g(x)\) \(f(g(x))\)
| x | g(x) | (f∘g)(x) |
|---|
Quizzes
Downloadable Review
Full set of practice questions for this unit.
Quick Practice Scratchpad
Try: Describe the transformation from \(f(x)=|x|\) to \(g(x)=-2\,|3(x-1)|+4\). Build a function \(f\) so that \(f^{-1}(x)=2x-5\) on a suitable domain.