Chapter 1

In Exercises 11 through 14 , find the center and radius of each circle, and draw a sketch of the graph. $$ 2 x^{2}+2 y^{2}-2 x+2 y+7=0 $$

In Exercises 1 through 4, find an equation of the circle with center at \(C\) and radius \(r\). Write the equation in both the centerradius form and the general form. $$ C(4,-3), r=5 $$

In Exercises 1 through 10 , list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ A \cup B $$

In Exercises 1 through 10, solve for \(x\). $$ |4 x+3|=7 $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(1,-2) $$

In Exercises 1 through 4 , find the slope of the line through the given points. $$ (2,-3),(-4,3) $$

Given \(g(x)=3 x^{2}-4\), find: (a) \(g(-4)\) (b) \(g\left(\frac{1}{2}\right)\) (c) \(g\left(x^{2}\right)\) (d) \(g\left(3 x^{2}-4\right)\) (e) \(g(x-h)\) (f) \(g(x)-g(h)\) (g) \(\frac{g(x+h)-g(x)}{h}, h \neq 0\)

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ g=\left\\{(x, y) \mid y=x^{2}+2\right\\} $$

In Exercises 1 through 4, find an equation of the circle with center at \(C\) and radius \(r\). Write the equation in both the centerradius form and the general form. $$ C(0,0), r=8 $$

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ C \cup D $$

In Exercises 1 through 10, solve for \(x\). $$ |3 x-8|=4 $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(-2,2) $$

In Exercises 1 through 4 , find the slope of the line through the given points. $$ (5,2),(-2,-3) $$

Given \(F(x)=\sqrt{2 x+3}\), find: (a) \(F(-1)\) (b) \(F(4)\) (c) \(F\left(\frac{t}{2}\right)\) (d) \(F(30)\) (e) \(F(2 x+3)\) (f) \(\frac{F(x+h)-F(x)}{h}, h \neq 0\)

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ F=\left\\{(x, y) \mid y=3 x^{2}-6\right\\} $$

In Exercises 1 through 4, find an equation of the circle with center at \(C\) and radius \(r\). Write the equation in both the centerradius form and the general form. $$ C(-5,-12), r=3 $$

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ A \cap B $$

In Exercises 1 through 10, solve for \(x\). $$ |5-2 x|=11 $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(2,2) $$

In Exercises 1 through 4 , find the slope of the line through the given points. $$ \left(\frac{1}{3}, \frac{1}{2}\right),\left(-\frac{5}{6}, \frac{2}{3}\right) $$

Given \(G(x)=\sqrt{2 x^{2}+1}\), find: (a) \(G(-2)\) (b) \(G(0)\) (c) \(G\left(\frac{1}{b}\right)\) (d) \(G\left(\frac{4}{7}\right)\) (e) \(G\left(2 x^{2}-1\right)\) (f) \(\frac{G(x+h)-G(x)}{h}, h \neq 0\)

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ G=\\{(x, y) \mid y=\sqrt{x+1}\\} $$

In Exercises 1 through 4, find an equation of the circle with center at \(C\) and radius \(r\). Write the equation in both the centerradius form and the general form. $$ C(-1,1), r=2 $$

In Exercises 1 through 10 , list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ C \cap D $$

In Exercises 1 through 10, solve for \(x\). $$ |4+3 x|=1 $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(-2,-2) $$

Derive midpoint formulas (2) and (3) if \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right)\) are both in the second quadrant and \(x_{2}>x_{1}\) and \(y_{1}>y_{2}\).

In Exercises 1 through 4 , find the slope of the line through the given points. $$ (-2.1,0.3),(2.3,1.4) $$

Given $$ f(x)= \begin{cases}\frac{|x|}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases} $$ find: (a) \(f(1) ;\) (b) \(f(-1) ;\) (c) \(f(4) ;\) (d) \(f(-4) ;\left(\right.\) e) \(f(-x)\); (f) \(f(x+1) ;\) (g) \(f\left(x^{2}\right) ;\left(\right.\) h) \(f\left(-x^{2}\right)\).

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ h=\\{(x, y) \mid y=\sqrt{3 x-4}\\} $$

In Exercises 5 through 10, find an equation of the circle satisfying the given conditions. Center is at \((1,2)\) and through the point \((3,-1)\).

In Exercises 1 through 10 , list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ B \cup D $$

In Exercises 1 through 10, solve for \(x\). $$ |5 x-3|=|3 x+5| $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(-1,-3) $$

Find the length of the medians of the triangle having vertices \(A(2,3), B(3,-3)\), and \(C(-1,-1)\).

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { The slope is } 4 \text { and through the point }(2,-3) \text {. } $$

Given \(f(t)=\frac{|3+t|-|t|-3}{t}\) express \(f(t)\) without absolute-value bars if (a) \(t>0\); (b) \(-3 \leq t<0 ;\) (c) \(t<-3\)

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ f=\left\\{(x, y) \mid y=\sqrt{4-x^{2}}\right\\} $$

In Exercises 5 through 10, find an equation of the circle satisfying the given conditions. Center is at \((-2,5)\) and tangent to the line \(x=7\).

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ A \cup C $$

In Exercises 1 through 10, solve for \(x\). $$ |x-2|=|3-2 x| $$

In Exercises 1 through 6, plot the given point \(P\) and such of the following points as may apply: (a) The point \(Q\) such that the line through \(Q\) and \(P\) is perpendicular to the \(x\) axis and is bisected by it. Give the coordinates of \(Q\). (b) The point \(R\) such that the line through \(P\) and \(R\) is perpendicular to and is bisected by the \(y\) axis. Give the coordinates of \(R\). (c) The point \(S\) such that the line through \(P\) and \(S\) is bisected by the origin. Give the coordinates of \(S\). (d) The point \(T\) such that the line through \(P\) and \(T\) is perpendicular to and is bisected by the \(45^{\circ}\) line through the origin bisecting the first and third quadrants. Give the coordinates of \(T\). $$ P(0,-3) $$

Find the midpoints of the diagonals of the quadrilateral whose vertices are \((0,0),(0,4),(3,5)\), and \((3,1)\).

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { Through the two points }(3,1) \text { and }(-5,4) \text {. } $$

In Exercises 7 through 12, the functions \(f\) and \(g\) are defined. In each problem define the following functions and determine the domain of the resulting function: (a) \(f+g ;\) (b) \(f-g ;\) (c) \(f \cdot g ;\) (d) \(f / g ;\) (e) \(g / f\); (f) \(f \circ g ;(\mathrm{g}) g \circ f\). $$ f(x)=x-5 ; g(x)=x^{2}-1 $$

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\). $$ B \cap D $$

Prove that the triangle with vertices \(A(3,-6), B(8,-2)\), and \(C(-1,-1)\) is a right triangle. Find the area of the triangle. (HINT: Use the converse of the Pythagorean theorem.)

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { Through the point }(-3,-4) \text { and parallel to the } y \text { axis. } $$

In Exercises 7 through 12, the functions \(f\) and \(g\) are defined. In each problem define the following functions and determine the domain of the resulting function: (a) \(f+g ;\) (b) \(f-g ;\) (c) \(f \cdot g ;\) (d) \(f / g ;\) (e) \(g / f\); (f) \(f \circ g ;(\mathrm{g}) g \circ f\). $$ f(x)=\sqrt{x} ; g(x)=x^{2}+1 $$

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ H=\\{(x, y)|y=| x-3 \mid\\} $$