Chapter 7

The graph of the rational function \(f\) is a hyperbola. The asymptotes of the graph of \(f\) intersect at \((3,2)\). The point \((2,1)\) is on the graph. Find another point on the graph. Explain your reasoning.

You are hired to wash the new cars at a car dealership with two other employees. You take an average of 40 minutes to wash a car \(\left(R_1=1 / 40\right.\) car per minute \()\). The second employee washes a car in \(x\) minutes. The third employee washes a car in \(x+10\) minutes. a. Write expressions for the rates that each employee can wash a car. b. Write a single expression \(R\) for the combined rate of cars washed per minute by the group. c. Evaluate your expression in part (b) when the second employee washes a car in 35 minutes. How many cars per hour does this represent? Explain your reasoning.

Write the prime factorization of the number. If the number is prime, then write prime. 91

Describe the intervals where the graph of \(y=\frac{a}{x}\) is increasing or decreasing when (a) \(a>0\) and (b) \(a<0\). Explain your reasoning.

Write the prime factorization of the number. If the number is prime, then write prime. 72

Is it possible to write a rational equation that has the following number of solutions? Justify your answers. a. no solution b. exactly one solution c. exactly two solutions d. infinitely many solutions

Find the next two expressions in the pattern shown. Then simplify all five expressions. What value do the expressions approach? $$ 1+\frac{1}{2+\frac{1}{2}}, 1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}, \ldots $$

Write the prime factorization of the number. If the number is prime, then write prime. 79

Let \(a\) be a nonzero real number. Tell whether each statement is always true, sometimes true, or never true. Explain your reasoning. a. For the equation \(\frac{1}{x-a}=\frac{x}{x-a}, x=a\) is an extraneous solution. b. The equation \(\frac{3}{x-a}=\frac{x}{x-a}\) has exactly one solution. c. The equation \(\frac{1}{x-a}=\frac{2}{x+a}+\frac{2 a}{x^2-a^2}\) has no solution.

The Doppler effect occurs when the source of a sound is moving relative to a listener, so that the frequency \(f_{\ell}\) (in hertz) heard by the listener is different from the frequency \(f_s\) (in hertz) at the source. In both equations below, \(r\) is the speed (in miles per hour) of the sound source. Moving away: Approaching: $$ f_{\ell}=\frac{740 f_s}{740+r} \quad f_{\ell}=\frac{740 f_s}{740-r} $$ a. An ambulance siren has a frequency of 2000 hertz. Write two equations modeling the frequencies heard when the ambulance is approaching and when the ambulance is moving away. b. Graph the equations in part (a) using the domain \(0 \leq r \leq 60\). c. For any speed \(r\), how does the frequency heard for an approaching sound source compare with the frequency heard when the source moves away?

Solve the system by graphing. \(y=x^2+6\) \(y=3 x+4\)

Your friend claims that it is not possible for a rational equation of the form \(\frac{x-a}{b}=\frac{x-c}{d}\), where \(b \neq 0\) and \(d \neq 0\), to have extraneous solutions. Is your friend correct? Explain your reasoning.

Factor the polynomial. $$ 4 x^2-4 x-80 $$

Solve the system by graphing. \(2 x^2-3 x-y=0\) \(\frac{5}{2} x-y=\frac{9}{4}\)

Is the domain discrete or continuous? Explain. Graph the function using its domain. The linear function \(y=0.25 x\) represents the amount of money \(y\) (in dollars) of \(x\) quarters in your pocket. You have a maximum of eight quarters in your pocket.

Factor the polynomial. $$ 3 x^2-3 x-6 $$

Solve the system by graphing. \(3=y-x^2-x\) \(y=-x^2-3 x-5\)

Factor the polynomial. $$ 2 x^2-2 x-12 $$

Solve the system by graphing. \(y=(x+2)^2-3\) \(y=x^2+4 x+5\)

Evaluate the function for the given value of \(x\). $$f(x)=x^3-2 x+7 ; x=-2$$

Factor the polynomial. $$ 10 x^2+31 x-14 $$

Evaluate the function for the given value of \(x\). $$g(x)=-2 x^4+7 x^3+x-2 ; x=3$$

Simplify the expression. $$ 3^2 \cdot 3^4 $$

Evaluate the function for the given value of \(x\). $$h(x)=-x^3+3 x^2+5 x ; x=3$$

Simplify the expression. $$ 2^{1 / 2} \cdot 2^{3 / 5} $$

Evaluate the function for the given value of \(x\). $$k(x)=-2 x^3-4 x^2+12 x-5 ; x=-5$$

Simplify the expression. $$ \frac{6^{5 / 6}}{6^{1 / 6}} $$

Simplify the expression. $$ \frac{6^8}{6^{10}} $$