10a.\(f(x) = x - 4\)
\(h(x) = \sqrt{x - 5}\)
\((f - h)(6) = (6 - 4) - (\sqrt{6 - 5})\)
\((f - h)(6) = 2 - \sqrt{1}\)
\((f - h)(6) = 2 - 1\)
\((f - h)(6) = 1\)
10b.\(f(x) = x - 4\)
\(g(x) = \frac{1}{x - 3}\)
\((f * g)(x) = (x - 4)(\frac{1}{x - 3})\)
\((f * g)(x) = \frac{x - 4}{x - 3}\)
Domain: (-∞, 3) ∨ (3, ∞) {x|x ≠ 3}
Interval Notation: (3 < x < 3), (3 > x > 3)
10c.\(g(x) = \frac{1}{x - 3}\)
\(h(x) = \sqrt{x - 5}\)
\((g * h)(x) = (\frac{1}{x - 3})(\sqrt{x - 5})\)
\((g * h)(x) = \frac{\sqrt{x - 5}}{x - 3}\)
10d.\(f(x) = x - 4\)
\(g(x) = \frac{1}{x - 3}\)
\(h(x) = \sqrt[3]{x - 7}\)
\(h(x) = (f * g)(x)\)
\(\sqrt[3]{x - 7} = (x - 4)(\frac{1}{x - 3})\)
\(\sqrt[3]{x - 7} = \frac{x - 4}{x - 3}\)
\((\sqrt[3]{x - 7})^{3} = (\frac{x - 4}{x - 3})^{3}\)
\(x - 7 = \frac{(x - 4)^{3}}{(x - 3)^{3}}\)
\((x - 3)^{3}(x - 7) = (x - 4)^{3}\)
\((x^{3} - 9x^2 + 27x + 27)(x - 7) = (x^{3} - 12x^{2} + 48x - 64)\)
\(x^{4} - 16x^{3} + 90x^{2} - 216x + 189 = x^{3} - 12x^{2} + 48x - 64\)
\(x^{4} + 90x^{2} - 216x + 189 = 17x^{3} - 12x^{2} + 48x - 64\)
\(x^{4} + 102x^{2} - 216x + 189 = 17x^{3} + 48x - 64\)
\(x^{4} + 102x^{2} + 189 = 17x^{3} + 264x - 64\)
\(x^{4} + 102x^{2} + 253 = 17x^{3} + 264x\)
\(x^{4} - 17x^{3} + 102x^{2} - 264x + 253 = 0\)
\(x = 4\ or\ 8\)
