Chapter 7

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(3 \sqrt{3-x}=10\)

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=2 x^{2}+2 $$

Solve. Check for extraneous solutions. \((3 x+2)^{\frac{1}{2}}-(2 x+7)^{\frac{1}{2}}=0\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ h)(-1) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \sqrt{72}+\sqrt{32}+\sqrt{18} $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt{x}}{\sqrt{2}} $$

The optimal height \(h\) of the letters of a message printed on pavement is given by the formula \(h=\frac{0.0052 d^{227}}{e} .\) Here \(d\) is the distance of the driver from the letters and \(e\) is the height of the driver's eye above the pavement. All of the distances are in meters. Find \(h\) for the given values of \(d\) and \(e .\) $$d=50 \mathrm{m}, e=1.2 \mathrm{m}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[4]{x^{8} y^{12}} $$

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(2.5 \sqrt{2 x-1.3}=-1\)

Solve. Check for extraneous solutions. \(x+8=\left(x^{2}+16\right)^{\frac{1}{2}}\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ g)(3) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \sqrt{75}+2 \sqrt{48}-5 \sqrt{3} $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt{5}}{\sqrt{8 x}} $$

The optimal height \(h\) of the letters of a message printed on pavement is given by the formula \(h=\frac{0.0052 d^{227}}{e} .\) Here \(d\) is the distance of the driver from the letters and \(e\) is the height of the driver's eye above the pavement. All of the distances are in meters. Find \(h\) for the given values of \(d\) and \(e .\) $$d=50 \mathrm{m}, e=2.3 \mathrm{m}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{32 y^{10}} $$

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(2 \sqrt{x+4}=3 \sqrt{x-1}\)

The formula for converting from Celsius to Fahrenheit temperatures is \(C=\frac{9}{5} F+32 .\) a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the Fahrenheit temperature that corresponds to \(25^{\circ} \mathrm{C} .\)

Solve. Check for extraneous solutions. \((2 x)^{\frac{1}{2}}=(x+5)^{\frac{1}{2}}\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ h)(2) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ 5 \sqrt{32 x}+4 \sqrt{98 x} $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt[3]{x}}{\sqrt[3]{2}} $$

The optimal height \(h\) of the letters of a message printed on pavement is given by the formula \(h=\frac{0.0052 d^{227}}{e} .\) Here \(d\) is the distance of the driver from the letters and \(e\) is the height of the driver's eye above the pavement. All of the distances are in meters. Find \(h\) for the given values of \(d\) and \(e .\) $$d=25 \mathrm{m}, e=2.3 \mathrm{m}$$

Geometry The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3} .\) Find the radius to the nearest hundredth of a sphere with each volume. $$ 10 \mathrm{in} .^{3} $$

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(\sqrt{2 x+5}=\sqrt{2-x}\)

Geometry The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3} .\) a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the radius of a sphere that has a volume of \(35,000 \mathrm{ft}^{3}\) .

Solve. Check for extraneous solutions. \(1=(3+x)^{\frac{1}{2}}\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ h)(-4) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \sqrt{75}-4 \sqrt{18}+2 \sqrt{32} $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \sqrt[3]{\frac{5}{3 x}} $$

Simplify each number. $$8^{\frac{2}{3}}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{9 x-9}\)

a. Form a pair of simultaneous equations by letting \(y_{1}\) equal the left side and \(y_{2}\) equal the right side of \(\sqrt{5}-x=1\) . Graph the equations. b. Repeat part (a) with the equivalent equation \(\sqrt{5}=x+1\) c. Repeat part (a) with the equivalent equation \(\sqrt{5}-x-1=0\) d. Writing Describe the similarities and differences among the graphs of the three sets of simultaneous equations.

For Exercises \(31-34, f(x)=10 x-10 .\) Find each value. $$ \left(f^{-1} \circ f\right)(10) $$

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(-2) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ 4 \sqrt{216 y^{2}}+3 \sqrt{54 y^{2}} $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt[4]{2}}{\sqrt[4]{5}} $$

Simplify each number. $$64^{\frac{2}{3}}$$

Geometry The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3} .\) Find the radius to the nearest hundredth of a sphere with each volume. $$ 0.45 \mathrm{cm}^{3} $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=-\sqrt{16 x+32}\)

For Exercises \(31-34, f(x)=10 x-10 .\) Find each value. $$ \left(f \circ f^{-1}\right)(-10) $$

a. Package Design The formula for the area \(A\) of a hexagon with a side \(s\) units long is \(A=\frac{3 s^{2} \sqrt{3}}{2}\) . See the figure below. Solve the formula for \(s\) and rationalize the denominator. b. A package designer wants the hexagonal base of a hat box to have an area of about 200 in. About how long is each side? c. What is the distance between opposite sides of the hat box?

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(-2) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ 3 \sqrt[3]{16}-4 \sqrt[3]{54}+\sqrt[3]{128} $$

$$ \frac{15 \sqrt{60 x^{5}}}{3 \sqrt{12 x}} $$

Simplify each number. $$(-8)^{\frac{2}{3}}$$

Geometry The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3} .\) Find the radius to the nearest hundredth of a sphere with each volume. $$ 0.002 \mathrm{mm}^{3} $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=-2 \sqrt{49 x+49}\)

For Exercises \(31-34, f(x)=10 x-10 .\) Find each value. $$ \left(f^{-1} \circ f\right)(0.2) $$

Your can find the area \(A\) of a square whose side is \(s\) units as \(A=s^{2} .\) Find the best estimate for the side of a square with an area of 32 \(\mathrm{m}^{2}\). A. 4.2 \(\mathrm{m}\) B. 5.7 \(\mathrm{m}\) C. 8.0 \(\mathrm{m}\) D. 16 \(\mathrm{m}\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(0) $$