1. $ f(x) = \sqrt{x - 2} $, $ g(x) = 2x^2 $. Find $ f \circ g $.
- $ f \circ g = f(g(x)) = f(2x^2) = \sqrt{2x^2 - 2} $
- Correct answer: A
2. $ f(x) = \frac{x^2 + 2}{x} $, $ g(x) = \frac{5}{1 - 3x^2} $, $ h(x) = \sqrt{x - 1} $. Find $ g \circ h \circ f $.
- $ h \circ f = h(f(x)) = h\left(\frac{x^2 + 2}{x}\right) = \sqrt{\frac{x^2 + 2}{x} - 1} = \sqrt{\frac{x^2 + 2 - x}{x}} = \sqrt{\frac{x^2 - x + 2}{x}} $
- $ g \circ h \circ f = g(h(f(x))) = g\left(\sqrt{\frac{x^2 - x + 2}{x}}\right) = \frac{5}{1 - 3\left(\sqrt{\frac{x^2 - x + 2}{x}}\right)^2} = \frac{5}{1 - 3\left(\frac{x^2 - x + 2}{x}\right)} = \frac{5}{\frac{x - 3(x^2 - x + 2)}{x}} = \frac{5x}{x - 3x^2 + 3x - 6} = \frac{5x}{-3x^2 + 4x - 6} $
- Correct answer: B
3. $ f(x) = \frac{8}{x} $, $ f \cdot g(x) = \frac{4}{x - 3} $. Find $ g(x) $.
- $ f(x) \cdot g(x) = \frac{8}{x} \cdot g(x) = \frac{4}{x - 3} $
- $ g(x) = \frac{4}{x - 3} \cdot \frac{x}{8} = \frac{4x}{8(x - 3)} = \frac{x}{2(x - 3)} $
- Correct answer: A
4. $ f(x) = x - 1 $, $ g'(x) = x^2 + 2x $. Find $ g(x) $.
- $ g'(x) = x^2 + 2x $
- $ g(x) = \int (x^2 + 2x) \, dx = \frac{x^3}{3} + x^2 + C $
- Since $ f(x) = x - 1 $, and $ g'(x) $ is given, we need to find $ g(x) $ such that $ g'(x) = x^2 + 2x $
- $ g(x) = \frac{x^3}{3} + x^2 + C $
- Correct answer: B
5. $ h(x) = 5x - 3 $, $ k \cdot h(x) = \frac{2x - 1}{x + 1} $. Find $ k(x) $.
- $ k(x) \cdot h(x) = \frac{2x - 1}{x + 1} $
- $ k(x) = \frac{2x - 1}{x + 1} \cdot \frac{1}{5x - 3} = \frac{2x - 1}{(x + 1)(5x - 3)} $
- Correct answer: B
6. $ f(x) = x - 4 $, $ g(x) = x^2 $. Find $ h $ such that $ h \circ g \circ f = x^2 - 8x + 3 $.
- $ g \circ f = g(f(x)) = g(x - 4) = (x - 4)^2 = x^2 - 8x + 16 $
- $ h(g \circ f) = h(x^2 - 8x + 16) = x^2 - 8x + 3 $
- Let $ u = x^2 - 8x + 16 $, then $ h(u) = u - 13 $
- $ h(x) = x - 13 $
- Correct answer: B
Parent Tip: Review the logic above to help your child master the concept of composition of functions worksheet.
