Chapter 5

For the following exercises, determine the domain and range of the quadratic function. $$ f(x)=-2(x+3)^{2}-6 $$

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as the square root of \(w\) and the square of \(t\). When \(x=3, z=1, w=25\), and \(t=2,\) then \(y=6\).

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{x^{2}+x+6}{x^{2}-10 x+24} $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{3}{x-4} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ 2 x^{3}+7 x^{2}-10 x-24=0 $$

For the following exercises, determine the end behavior of the functions. $$ f(x)=x^{2}\left(2 x^{3}-x+1\right) $$

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{3}-5 x^{2}+13\right) \div(x+4) $$

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{5}-5 x^{3}+4 x $$

Determine the end behavior of the functions. $$f(x)=x^{2}\left(2 x^{3}-x+1\right)$$

For the following exercises, determine the domain and range of the quadratic function. $$ f(x)=x^{2}+6 x+4 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as \(x\). When \(x=3\), then \(y=12\). Find \(y\) when \(x=20\).

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{94-2 x^{2}}{3 x^{2}-12} $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{x+3}{x+7} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ x^{3}+2 x^{2}-9 x-18=0 $$

For the following exercises, determine the end behavior of the functions. $$ f(x)=(2-x)^{7} $$

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{3}-3 x+2\right) \div(x+2) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. $$ f(x)=x^{3}-9 x, \text { between } x=-4 \text { and } x=-2 $$

Determine the end behavior of the functions. $$f(x)=(2-x)^{7}$$

For the following exercises, determine the domain and range of the quadratic function. $$ f(x)=2 x^{2}-4 x+2 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the square of \(x\). When \(x=2\), then \(y=16 .\) Find \(y\) when \(x=8\).

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{x}{2 x+1} $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{x-2}{x+7} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ x^{3}+5 x^{2}-16 x-80=0 $$

For the following exercises, find the intercepts of the functions. $$ f(t)=2(t-1)(t+2)(t-3) $$

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{3}-21 x^{2}+147 x-343\right) \div(x-7) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=x^{3}-9 x,\) between \(x=2\) and \(x=4\)

Find the intercepts of the functions. $$f(t)=2(t-1)(t+2)(t-3)$$

For the following exercises, determine the domain and range of the quadratic function. $$ k(x)=3 x^{2}-6 x-9 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the cube of \(x\). When \(x=3\), then \(y=5 .\) Find \(y\) when \(x=4\).

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{2 x}{x-6} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ x^{3}-3 x^{2}-25 x+75=0 $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{3 x+4}{5-4 x} $$

For the following exercises, determine the end behavior of the functions. $$ g(n)=-2(3 n-1)(2 n+1) $$

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{3}-15 x^{2}+75 x-125\right) \div(x-5) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=x^{5}-2 x,\) between \(x=1\) and \(x=2\)

Find the intercepts of the functions. $$g(n)=-2(3 n-1)(2 n+1)$$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(2,0),(x, y)=(4,4) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the square root of \(x\). When \(x=16,\) then \(y=4\). Find \(y\) when \(x=36\).

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{-2 x}{x-6} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ 2 x^{3}-3 x^{2}-32 x-15=0 $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{5 x+1}{2-5 x} $$

For the following exercises, determine the end behavior of the functions. $$ f(x)=x^{4}-16 $$

For the following exercises, use synthetic division to find the quotient. $$ \left(9 x^{3}-x+2\right) \div(3 x-1) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=-x^{4}+4,\) between \(x=1\) and \(x=3\)

Find the intercepts of the functions. $$f(x)=x^{4}-16$$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(-2,-1),(x, y)=(-4,3) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the cube root of \(x\). When \(x=125,\) then \(y=15 .\) Find \(y\) when \(x=1,000 .\)

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{x^{2}-4 x+3}{x^{2}-4 x-5} $$

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ 2 x^{3}+x^{2}-7 x-6=0 $$

For the following exercises, find the inverse of the functions. $$ f(x)=x^{2}+2 x,[-1, \infty) $$