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Find the inverse of the function. (Hint: Try rewriting the function by using either inspection or long division.) $$f(x)=\frac{3 x+1}{x-4}$$

Find the inverse of the function. (Hint: Try rewriting the function by using either inspection or long division.) $$f(x)=\frac{4 x-7}{2 x+3}$$

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.

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

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?

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.