Chapter 5

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=\frac{1}{x^{2}}, \quad x \geq 0 $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ q(x)=\frac{x-5}{3 x-1} $$

For the following exercises, find all complex solutions (real and non-real). $$ x^{3}+x^{2}+x+1=0 $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x+3,-4 x^{3}+5 x^{2}+8 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=2 x^{4}\left(x^{3}-4 x^{2}+4 x\right) $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly with the square of \(x\) and when \(x=2, y=3\).

For the following exercises, use a graph to help determine the domain of the functions. $$ f(x)=\sqrt{\frac{(x+1)(x-1)}{x}} $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ s(x)=\frac{4}{(x-2)^{2}} $$

For the following exercises, find all complex solutions (real and non-real). $$ x^{3}-8 x^{2}+25 x-26=0 $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2, \quad 4 x^{4}-15 x^{2}-4 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{4}\left(9 x^{4}-12 x^{3}+4 x^{2}\right) $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the cube of \(x\) and when \(x=2, y=4\).

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ r(x)=\frac{5}{(x+1)^{2}} $$

For the following exercises, find all complex solutions (real and non-real). $$ x^{3}+13 x^{2}+57 x+85=0 $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-\frac{1}{2}, \quad 2 x^{4}-x^{3}+2 x-1 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ f(x)=(x+3)^{2}(x-2) $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=2 .\)

For the following exercises, use a graph to help determine the domain of the functions. $$ f(x)=\sqrt{\frac{x(x+3)}{x-4}} $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ f(x)=\frac{3 x^{2}-14 x-5}{3 x^{2}+8 x-16} $$

For the following exercises, find all complex solutions (real and non-real). $$ 3 x^{3}-4 x^{2}+11 x+10=0 $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x+\frac{1}{3}, \quad 3 x^{4}+x^{3}-3 x+1 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ g(x)=(x+4)(x-1)^{2} $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely with \(x\) and when \(x=6, y=2\).

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ g(x)=\frac{2 x^{2}+7 x-15}{3 x^{2}-14 x+15} $$

For the following exercises, find all complex solutions (real and non-real). $$ x^{4}+2 x^{3}+22 x^{2}+50 x-75=0 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ h(x)=(x-1)^{3}(x+3)^{2} $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely as the square of \(x\) and when \(x=1, y=4\).

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ a(x)=\frac{x^{2}+2 x-3}{x^{2}-1} $$

For the following exercises, find all complex solutions (real and non-real). $$ 2 x^{3}-3 x^{2}+32 x+17=0 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ k(x)=(x-3)^{3}(x-2)^{2} $$

For the following exercises, use Kepler's Law, which states that the square of the time, \(T,\) required for a planet to orbit the Sun varies directly with the cube of the mean distance, \(a\), that the planet is from the Sun. Using Earth's time of 1 year and mean distance of 93 million miles, find the equation relating \(T\) and \(a\).

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. $$ f(x)=x^{3}-x-2, y=1,2,3 $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ b(x)=\frac{x^{2}-x-6}{x^{2}-4} $$

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=x^{3}-1 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ m(x)=-2 x(x-1)(x+3) $$

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=-x^{3} $$

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 5 & 2 & 1 & 2 & 5 \\ \hline \end{array} $$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. $$ f(x)=x^{3}+x-2, y=0,1,2 $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ h(x)=\frac{2 x^{2}+x-1}{x-4} $$

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=x^{4}-x^{2}-1 $$

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ n(x)=-3 x(x+2)(x-4) $$

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{4}-5 x^{2} $$

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 0 & 1 & 4 & 9 \\ \hline \end{array} $$

For the following exercises, use Kepler's Law, which states that the square of the time, \(T,\) required for a planet to orbit the Sun varies directly with the cube of the mean distance, \(a\), that the planet is from the Sun. Using Earth's distance of 150 million kilometers, find the equation relating \(T\) and \(a\).

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. $$ f(x)=x^{3}+3 x-4, y=0,1,2 $$

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ k(x)=\frac{2 x^{2}-3 x-20}{x-5} $$

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=x^{3}-2 x^{2}-5 x+6 $$

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{2}(1-x)^{2} $$

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. $$ \begin{array}{cccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & -2 & 1 & 2 & 1 & -2 \end{array} $$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. $$ f(x)=x^{3}+8 x-4, y=-1,0,1 $$