Sec 2.1: Functions

  1. A piecewise defined function is given as \( f(x) = \begin{cases} x^2+4x & \text{if } x \le -1, \\ x & \text{if } -1 < x \le 1, \\ -1 & \text{if } x > 1. \end{cases}\) Evaluate \( f(x)\) at the indicated values.

    2. (a) \( f(-2) \)  (b) \( f(0) \)  (c) \( f(-1) \)   (d) \(f(-5/4)\)   (e) \(f(30)\)

  2. Evaluate the function \( f(x) = x^2-7x+8\) at the given values of the independent variable and simplify.   solution

    3. (a) \( f(-2) \)    (b) \( f(x+1) \)   (c) \( f(-x) \)

  3. If \( f(x)=\begin{cases} 2x-7,   \quad  x  \geq 0, \\  \sqrt{5-2x},  \quad  x  < 0,     \end{cases} \)  and  \(g(x)=\begin{cases}   x^2-x+1,  \quad x < 1, \\  |2x-3| , \quad   x \geq 1,  \end{cases} \) evaluate the following:  solution

    4. (a) \(f(-2)\)     (b) \(g(-3)\)       (c) \(g(1)\)        (d) \(f(5)\)       (e)  \(g(0)\)        (f) \(g(3)\)

     

  4. Suppose the tax liability \(T \) on \(x\) dollars of taxable income is given by

    5. \(T(x)=\begin{cases}
    0.06x \qquad \qquad \quad \text{if}  \, \, 0 < x < 30,000,\\
    1,800+0.04x \qquad \text{if}  \, \, 30,000 \leq x < 80,000,\\
    5,000+0.05x \qquad \text{if} \,  \, x \geq 80,000.
    \end{cases}\)  

    Find the tax liability on each taxable income.   solution

    (a) \(\displaystyle{\$25,000}\)            (b) \(\displaystyle{\$125,000}\)            (c) \(\displaystyle{\$60,000}\)       

  5. Evaluate the difference quotient as indicated.

    6. (a) \( \dfrac{f(a+h)-f(a)}{h}\)   for   \( f(x)=x^2\)  solution

    (b) \( \dfrac{f(4+h)-f(4)}{h}\)   for   \( f(x)=\sqrt{x}\)  solution

    (c) \( \dfrac{f(x+h)-f(x)}{h}\)   for   \( f(x)=\dfrac{1}{\sqrt{x}}\)  solution

  6. Given \( f(x)=5x-7 \). Evaluate \(\dfrac{f(1+h)-f(1)}{h} \).  solution

  7. Find the difference quotient \(\dfrac{f(4+h)-f(4)}{h} \) and simplify for the function \( f(x)=\sqrt{x}\).   solution

  8. Find the difference quotient \(\dfrac{f(a+h)-f(a)}{h}\) and simplify for the function \(f(x)=3x^2-5x \).  solution

  9. Find the domain of the following functions in interval notation.   solution

    10. (a)   \( \displaystyle{f(x)=\frac{3}{x+2}} \)  (b)   \(\displaystyle{f(x)=\frac{x^2+9}{x^2-9}}\)  (c)  \(\displaystyle{f(x)=x^2-x+1}\)   (d) \( \displaystyle{ f(x)= \frac{2}{\sqrt{4-x^2}} }\) (e)   \(\displaystyle{ f(x)= \frac{\sqrt{4-2x}}{x+5}}\)

  10. Find the domain of the function.   solution

    11. (a) \(g(x)=\sqrt{x+5} \)   (b) \(f(x)=\dfrac{x-5}{\sqrt{43+8x}} \)