Chapter 2

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(2 y-x y+3 x=4\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=2 x-3, \quad g(x)=2 x-3\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\frac{1}{2} \sqrt[3]{x}-3\)

Evaluate the function at each specified value of the independent variable. . \(f(x)=\llbracket-x \rrbracket\) (a) \(f(3)\) (b) \(f(6.1)\) (c) \(f(-5.9)\) (d) \(f(-9)\)

A salesperson receives a monthly salary of $$\$ 2500$$ plus a commission of \(7 \%\) of sales. Write a linear equation for the salesperson's monthly wage \(W\) in terms of the person's monthly sales \(S .\)

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\begin{array}{ll}\underline{\phantom{xxx}}(3,-2) & m \text { is undefined. }\end{array}\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(x^{2} y-x^{2}+4 y=0\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=|x|, \quad g(x)=x+6\)

Evaluate the function at each specified value of the independent variable. \(f(x)=\llbracket x-1.8 \rrbracket\) (a) \(f(4)\) (b) \(f(3.7)\) (c) \(f(-5.8)\) (d) \(f(-6.3)\)

A forest region had a population of 1300 deer in the year 2000 . During the next 8 years, the deer population increased by about 60 deer per year. (a) Write a linear equation giving the deer population \(P\) in terms of the year \(t\). Let \(t=0\) represent 2000 . (b) The deer population keeps growing at this constant rate. Predict the number of deer in 2012 .

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-2,-7) \quad m=0\)

Use your knowledge of the Cartesian plane and intercepts to explain why you let \(y\) equal zero when you are finding the \(x\) -intercepts of the graph of an equation, and why you let \(x\) equal zero when you are finding the \(y\) -intercepts of the graph of an equation.

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=x^{2 / 3}, \quad g(x)=x^{6}\)

Evaluate the function at each specified value of the independent variable and simplify. \(A(s)=\frac{\sqrt{3} s^{2}}{4}\) (a) \(A(1)\) (b) \(A(0)\) (c) \(A(2 x)\) (d) \(A(3)\)

The cost of implementing an invasive species management system in a forest is related to the area of the forest. It costs \(\$ 630\) to implement the system in a forest area of 10 acres. It costs \(\$ 1070\) in a forest area of 18 acres. (a) Write a linear equation giving the cost of the invasive species management system in terms of the number of acres \(x\) of forest. (b) Use the equation in part (a) to find the cost of implementing the system in a forest area of 30 acres.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-10,4)\) \(m=0\)

Is it possible for a graph to have no \(x\) -intercepts? no. \(y\) -intercepts? no \(x\) -intercepts and no \(y\) -intercepts? Give examples to support your answers.

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=x^{2}+3, \quad g(x)=\sqrt{x}\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=3\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(y)=3-\sqrt{y}\) (a) \(f(4)\) (b) \(f(100)\) (c) \(f\left(4 x^{2}\right)\) (d) \(f(0.25)\)

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\left(4, \frac{5}{2}\right) \quad m=\frac{4}{3}\)

Check for symmetry with respect to both axes and the origin. \(x^{4}-2 y=0\)

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\sqrt[3]{x+1}, \quad g(x)=x^{3}\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(g(x)=x\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=\sqrt{x+3}-2\) (a) \(f(-3)\) (b) \(f(1)\) (c) \(f(x-3)\) (d) \(f(x+4)\)

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\left.\left(-\frac{1}{2}, \frac{3}{2}\right)\right) \quad m=-3\)

Check for symmetry with respect to both axes and the origin. \(y=x^{4}-x^{2}+3\)

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\frac{1}{x^{2}}, \quad g(x)=x-2\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=5-3 x\)

Evaluate the function at each specified value of the independent variable and simplify. \(c(x)=\frac{1}{x^{2}-16}\) (a) \(c(4)\) (b) \(c(0)\) (c) \(c(y+2)\) (d) \(c(y-2)\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(y=2 x-1\)

Check for symmetry with respect to both axes and the origin. \(x-y^{2}=0\)

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\frac{5}{x^{2}-4}, \quad g(x)=x+3\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(h(x)=x^{2}-4\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(y=3-x\)

Check for symmetry with respect to both axes and the origin. \(y^{2}=x+2\)

Use a graphing utility to graph \(f\) for \(c=-2,0\), and 2 in the same viewing window. (a) \(f(x)=\frac{1}{2} x+c\) (b) \(f(x)=\frac{1}{2}(x-c)\) (c) \(f(x)=\frac{1}{2}(c x)\) In each case, compare the graph with the graph of \(y=\frac{1}{2} x\).

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(g(s)=\frac{s^{3}}{4}\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=\frac{|x|}{x}\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f\left(x^{2}\right)\) (d) \(f(x-1)\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(4 x-y-6=0\)

Check for symmetry with respect to both axes and the origin. \(y=\sqrt{16-x^{2}}\)

Use a graphing utility to graph \(f\) for \(c=-2,0\), and 2 in the same viewing window. (a) \(f(x)=x^{3}+c\) (b) \(f(x)=(x-c)^{3}\) (c) \(f(x)=(x-2)^{3}+c\) In each case, compare the graph with the graph of \(y=x^{3}\).

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(t)=-t^{4}\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=|x|+4\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f\left(x^{2}\right)\) (d) \(f(x+2)\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(2 x+3 y-9=0\)

Check for symmetry with respect to both axes and the origin. \(y=\sqrt{4-x^{2}}\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\sqrt{1-x}\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=\left\\{\begin{array}{ll}3 x-1, & x<0 \\ 2 x+3, & x \geq 0\end{array}\right.\) (a) \(f(-1)\) (b) \(f(0)\) (c) \(f(-2)\) (d) \(f(2)\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(8-3 x=0\)

Check for symmetry with respect to both axes and the origin. \(x y=2\)