Chapter 1

In Problems 1-22, sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=x^{2}+1 $$

State the range for the given functions. Graph each function. $$ f(x)=x^{2}, x \in \mathbf{R} $$

Find the two numbers that have distance 3 from \(-1\) by (a) measuring the distances on the real-number line and (b) solving an appropriate equation involving an absolute value.

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-(x-2)^{2}+1 $$

State the range for the given functions. Graph each function. $$ f(x)=x^{2}, x \in[0,1] $$

Find all pairwise distances between the numbers \(-5,2\), and 7 by (a) measuring the distances on the real-number line and (b) computing the distances by using absolute values.

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=x^{3}-2 $$

State the range for the given functions. Graph each function. $$ f(x)=x^{2},-1

Solve the following equations: (a) \(|2 x-4|=6\) (b) \(|x-3|=2\) (c) \(|2 x+3|=5\) (d) \(|7-3 x|=-2\)

State the range for the given functions. Graph each function. $$ f(x)=x^{2},-\frac{1}{2}

Solve the following equations: (a) \(|2 x+4|=|5 x-2|\) (b) \(|5-3 u|=|3+2 u|\) (c) \(\left|4+\frac{t}{2}\right|=\left|\frac{3}{2} t-2\right|\) (d) \(|2 s-3|=|7-s|\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-2 x^{2}-3 $$

(a) Show that, for \(x \neq 1\), $$ \frac{x^{2}-1}{x-1}=x+1 $$ (b) Are the functions $$ f(x)=\frac{x^{2}-1}{x-1}, \quad x \neq 1 $$ and $$ g(x)=x+1, \quad x \in \mathbf{R} $$ equal?

Solve the following inequalities: (a) \(|5 x-2| \leq 4\) (b) \(|1-3 x|>8\) (c) \(|7 x+4| \geq 3\) (d) \(|6-5 x|<7\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-(3-x)^{2} $$

(a) Show that $$ 2|x-1|=\left\\{\begin{array}{ll} 2(x-1) & \text { for } x \geq 1 \\ 2(1-x) & \text { for } x \leq 1 \end{array}\right. $$ (b) Are the functions $$ f(x)=\left\\{\begin{array}{ll} 2-2 x & \text { for } 0 \leq x \leq 1 \\ 2 x-2 & \text { for } 1 \leq x \leq 2 \end{array}\right. $$ and $$ g(x)=2|x-1|, x \in[0,2] $$ equal?

Solve the following inequalities: (a) \(|2 x+3|<6\) (b) \(|3-4 x| \geq 2\) (c) \(|x+5| \leq 1\) (d) \(|7-2 x|<0\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=3+1 / x $$

In Problems \(7-12\), sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. $$ f(x)=2 x $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((2,4)\) with slope \(-\frac{1}{3}\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=1-1 / x $$

Sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. $$ f(x)=3 x^{2} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-2)\) with slope 2

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=1 /(x-1) $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((0,-2)\) with slope \(-3\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=1+1 /(x+2)^{2} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-3,5)\) with slope \(1 / 2\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\exp (x-2) $$

Sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. $$ f(x)=-|x| $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-2,-3)\) and \((1,4)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\exp (-x) $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-1,4)\) and \(\left(2,-\frac{1}{2}\right)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ v=e^{-(x+3)} $$

Suppose that $$ f(x)=x^{2}, \quad x \in \mathbf{R} $$ and $$ g(x)=3+x, \quad x \in \mathbf{R} $$ (a) Show that $$ (f \circ g)(x)=(3+x)^{2}, \quad x \in \mathbf{R} $$ (b) Show that $$ (g \circ f)(x)=3+x^{2}, \quad x \in \mathbf{R} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((0,4)\) and \((3,0)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=3 e^{2 x+1} $$

Suppose that $$ f(x)=x^{3}, \quad x \in \mathbf{R} $$ and $$ g(x)=1-x, \quad x \in \mathbf{R} $$ (a) Show that $$ (f \circ g)(x)=(1-x)^{3}, \quad x \in \mathbf{R} $$ (b) Show that $$ (g \circ f)(x)=1-x^{3}, \quad x \in \mathbf{R} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-1)\) and \((4,5)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\ln (x+1) $$

Suppose that $$ f(x)=1-x^{2}, \quad x \in \mathbf{R} $$ and $$ g(x)=2 x, \quad x \geq 0 $$ (a) Find $$ (f \circ g)(x) $$ together with its domain. (b) Find $$ (g \circ f)(x) $$ together with its domain.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The horizontal line through \(\left(3, \frac{3}{2}\right)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\ln (x-3) $$

Suppose that $$ f(x)=\frac{1}{x+1}, \quad x \neq-1 $$ and $$ g(x)=2 x^{2}, \quad x \in \mathbf{R} $$ \(\begin{array}{ll}\text { (a) Find }(f \circ g)(x) . & \text { (b) Find }(g \circ f)(x) \text { . }\end{array}\) In both (a) and (b), find the domain.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The horizontal line through \((0,-1)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-\ln (x-1)+1 $$

Suppose that $$ f(x)=3 x^{2}, \quad x \geq 3 $$ and $$ g(x)=\sqrt{x}, \quad x \geq 0 $$ Find \((f \circ g)(x)\) together with its domain.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The vertical line through \(\left(-1, \frac{7}{2}\right)\)

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-\ln (1-x) $$

Suppose that $$ f(x)=x^{4}, \quad x \geq 3 $$ and $$ g(x)=\sqrt{x+1}, \quad x \geq 3 $$ Find \((f \circ g)(x)\) together with its domain.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The vertical line through \((2,-3)\)