Tut 3: Functions, Piecewise Functions, Domains and Ranges of Functions

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5. If \(f(x) = 3x^2 - x + 2\), find:

\(f(2), f(a), f(-a), f(a + 1), 2f(a), f(a^2), [f(a)]^2\) and \(f(a + h)\).

6. Evaluate the difference quotient for the given function. Simplify your answer.

(a) \(f(x) = 4 + 3x - x^2, \frac{f(3 + h) - f(3)}{h}\)
(b) \(f(x) = x^3, \frac{f(a + h) - f(a)}{h}\)
(c) \(f(x) = \frac{1}{x}, \frac{f(x) - f(a)}{x - a}\)
(d) \(f(x) = \frac{x+3}{x+1}, \frac{f(x) - f(1)}{x - 1}\)

7. Find the domain of the function

(a) \(f(x) = \frac{x+4}{x^2-9}\)
(b) \(f(x) = \frac{2x^3-5}{x^2+x -6}\)
(c) \(f(x) = \sqrt[3]{2x - 1}\)
(d) \(g(t) = \sqrt{3 - t} - \sqrt{2 - t}\)
(e) \(h(x) = \frac{1}{\sqrt[4]{x^2-5x}}\)

8. Find the domain and the range and sketch the graph of the function

\(h(x)=\sqrt{4 - x^2}\)

9. Find the domain and sketch the graph of the function

(e) \(g(x) = \sqrt{x - 5}\)
(d) \( H(t) = \frac{9-t^2}{3-t} \)
(g) \( G(x) = \frac{3x + |x|}{x}\)
(i) \(f(x) = \left\lbrace \begin{array}{c} x + 2 \mbox{ if } x < 0 \\ 1 - x \mbox{ if } x \geq 0 \end{array} \right.\)

(l) \(f(x) = \left\lbrace \begin{array}{l} x + 9 \mbox{ if } x < -3, \\ -2x \mbox{ if } |x| \leq 3, \\ -2x \mbox{ if } x > 3. \end{array} \right. \)

11. Find a formula for the described function and state its domain.

(a) A rectangle has perimeter 20m. Express the area of the\\ rectangle as a function of the length of one its sides.
(b) A rectangle has area \(16m^2\). Express the perimeter of the rectangle as a function of the length of one of its sides.