Function Composition and Inverses: math test, Grade 12 – download the PDF

1 Compositions / 4 pts

Let \(f(x)=x^2-4x\) and \(g(x)=3x+1\).

1. Find \((f\circ g)(x)\) and simplify.
2. Find \((g\circ f)(x)\).
3. Compute \((f\circ g)(1)\).
4. Solve \((f\circ g)(x)=0\).

2 Domains of composites / 3 pts

Let \(f(x)=\sqrt{x-1}\) and \(g(x)=6-x^2\). Find \(f\circ g\) and \(g\circ f\) and the domain of each.

3 A rational inverse / 4 pts

Let \(f(x)=\dfrac{2x+5}{x-1}\) for \(x\neq1\).

1. Find \(f^{-1}(x)\).
2. Verify that \(f(f^{-1}(x))=x\).
3. State the domain and the range of \(f\).

4 A restricted inverse / 3 pts

Let \(f(x)=x^2-6x+4\) for \(x\ge3\). Find \(f^{-1}\) and its domain, then verify with \(f^{-1}(11)\).

5 Transforming a root graph / 3 pts

Let \(g(x)=2\sqrt{x+3}-4\).

1. Describe the transformations of \(y=\sqrt{x}\) that give \(g\), and state the domain and range.
2. Find the x-intercept and the y-intercept (exact value, then a decimal rounded to the hundredth; a calculator is allowed).

6 Symmetry and pieces / 3 pts

(a) Classify \(f(x)=x^5+3x\), \(g(x)=x^4-|x|\) and \(h(x)=x^2+2x+1\) as even, odd, or neither.

(b) Rewrite \(k(x)=|x-1|+x\) as a piecewise function.