2.5 Chapter Review
2.5.1 Chapter summary
A function assigns one output to each input. Domain and range identify allowable inputs and resulting outputs, while graphs reveal shape, change, intersections, and extrema. Parent functions support efficient graphing through transformations. Function arithmetic combines outputs, composition links successive relationships, and inverses reverse one-to-one functions. In applied work, domains, units, and physical meaning remain essential.
2.5.2 Common mistakes
- Substituting a negative or algebraic input without parentheses.
- Confusing codomain with range.
- Ignoring a physical restriction after finding an algebraic domain.
- Reversing the direction of a horizontal shift.
- Assuming that \(f\circ g=g\circ f\).
- Treating \(f^{-1}\) as a reciprocal.
2.5.3 Exercises
- For \(f(x)=x^2-5x+4\), find \(f(-2)\).
Check Your Work
\(f(-2)=4+10+4=18\). - Find the domain of \(g(x)=\sqrt{x+2}/(x-4)\).
Check Your Work
\([-2,4)\cup(4,\infty)\). - Find the average rate of change of \(f(x)=x^2+1\) from \(x=2\) to \(x=6\).
Check Your Work
\((37-5)/(6-2)=8\). - Describe the transformations from \(x^2\) to \(-2(x-3)^2+5\).
Check Your Work
Vertical stretch by 2, reflection across the \(x\)-axis, right 3, and up 5. - If \(f(x)=2x-1\) and \(g(x)=\sqrt{x}\), find \((g\circ f)(x)\) and its domain.
Check Your Work
\(\sqrt{2x-1}\) with domain \([1/2,\infty)\). - Determine whether \(x^3+4\) is one-to-one and find its inverse.
Check Your Work
It is one-to-one. The inverse is \(f^{-1}(x)=\sqrt[3]{x-4}\). - Tank volume is \(V(t)=24+1.8t\) m³. Interpret its intercept and slope.
Check Your Work
The tank begins with 24 m³ and gains 1.8 m³ per unit of time. - A circular basin radius grows as \(r(t)=5+0.02t\). Write area as a function of time.
Check Your Work
\(A(t)=\pi(5+0.02t)^2\).