Function Composition Practice

Composition of Functions from Tables and Graphs

• Table Evaluations:

  • Given discrete function values from a table:
  • $(f \circ g)(1) = f(g(1)) = f(0) = 5$
  • $(g \circ f)(2) = g(f(2)) = g(1) = 0$
  • $(g \circ g)(1) = g(g(1)) = g(0) = 1$

• Graphical Evaluations:   

  • $(g \circ f)(5) = g(f(5)) = g(1) = 4$
  • $(f \circ g)(2) = f(g(2)) = f(2) = -2$

Function Decomposition

• Decomposing Composite Functions $H(x) = (f \circ g)(x)$:
  • For $H(x) = (2x + 3)^4$: $g(x) = 2x + 3$ and $f(x) = x^4$
  • For $H(x) = (1 + x^2)^3$: $g(x) = x^2$ and $f(x) = (1 + x)^3$
  • For $H(x) = |2x^2 + 3|$: $g(x) = 2x^2 + 3$ and $f(x) = |x|$

Algebraic Combinations and Compositions

• Given Base Functions:

  • $f(x) = x^2$
  • $g(x) = \frac{1}{x}$
  • $h(x) = \sqrt{x}$
  • $k(x) = e^x$
  • $A(x) = \ln(x)$

• Expressions via Combinations and Compositions:

  • $e^{\sqrt{x}} = (k \circ h)(x)$
  • $x^2 e^x = f(x) \cdot k(x)$
  • $\frac{e^x}{x^2} = \frac{k(x)}{f(x)}$
  • $x^2 + \sqrt{x} = f(x) + h(x)$
  • $\ln\left(\frac{1}{x}\right) = (A \circ g)(x)$
  • $\ln(x) - e^x = A(x) - k(x)$

Compositions and Domain Restrictions

• Polynomial Compositions:

  • Given $f(x) = x^2$ and $g(x) = x^2 - 4$:
  • $(f \circ g)(x) = (x^2 - 4)^2 = x^4 - 8x^2 + 16$ with domain $(-\infty, \infty)$
  • $(g \circ f)(x) = (x^2)^2 - 4 = x^4 - 4$ with domain $(-\infty, \infty)$

• Rational Compositions:

  • Given $f(x) = \frac{1}{x + 3}$ and $g(x) = -\frac{2}{x}$:
  • $(f \circ g)(x) = \frac{x}{3x - 2}$ with domain $(-\infty, 0) \cup \left(0, \frac{2}{3}\right) \cup \left(\frac{2}{3}, \infty\right)$
  • $(g \circ f)(x) = -2(x + 3) = -2x - 6$ with domain $(-\infty, -3) \cup (-3, \infty)$