Exam 1: Linear Functions, Equations, and Inequalities

Use the figure to solve each equation or inequality. (a) $f(x)=g(x)$ (b) $f(x)<g(x)$ (c) $f(x) \geq g(x)$ (d) $y_{2}-y_{1}=0$

Suppose that an empty circular wading pool has a radius of 5 feet. During a storm, rain falling at a rate of 1.5 inches per hour begins to fill the pool. A small drain at the bottom of the pool is capable of draining 30 gallons of water per hour. (a) Determine the number of cubic inches of water falling into the pool in one hour. (Hint: Each hour a layer of water 1.5 inches thick falls into the pool.) (b) One gallon of water equals about 231 cubic inches. Write a formula for a function $g$ that computes the gallons of water landing in the pool in $x$ hours. (c) How many gallons of water land in the pool during a 2.5 hour storm? (d) Will the drain be able to keep up with the rainfall? If not, how many such drains would be needed?

Suppose that an empty circular wading pool has a radius of 5 feet. During a storm, rain falling at a rate of 0.6 inch per hour begins to fill the pool. A small drain at the bottom of the pool is capable of draining 35 gallons of water per hour. (a) Determine the number of cubic inches of water falling into the pool in one hour. (Hint: Each hour a layer of water 0.6 inch thick falls into the pool.) (b) One gallon of water equals about 231 cubic inches. Write a formula for a function $g$ that computes the gallons of water landing in the pool in $x$ hours. (c) How many gallons of water land in the pool during a 2 hour storm? (d) Will the drain be able to keep up with the rainfall? If not, how many such drains would be needed?

Use the figure to solve each equation or inequality. (a) $f(x)=g(x)$ (b) $f(x)<g(x)$ (c) $f(x) \geq g(x)$ (d) $y_{2}-y_{1}=0$

Find the equation of the line passing through the point $(-4,0)$ and (a) parallel to the line with equation $y=3 x+7$ . (b) perpendicular to the line with equation $\frac{1}{3} x-y=5$ .

Use the screen to solve the equation or inequality. Here the function $y_{1}=f(x)$ is a linear function defined over the domain of real numbers. (a) $y_{1}=0$ (b) $y_{1}<0$ (c) $y_{1}>0$ (d) $y_{1} \leq 0$

Find the $x$ - and $y$ -intercepts of the line whose standard form is $-3 x+2 y=9$ . What is the slope of this line?

For each of the functions, determine the (i) domain (ii) range (iii) x-intercept(s) (iv) y-intercept(s). (a) (b) (c)

Use the figure to solve each equation or inequality. (a) $f(x)=g(x)$ (b) $f(x)<g(x)$ (c) $f(x) \geq g(x)$ (d) $y_{2}-y_{1}=0$

Consider the linear functions $f(x)=2(x-1)+7$ and $g(x)=x+3-3(x-2)$ . (a) Solve $f(x)=g(x)$ analytically, showing all steps. Also, check analytically. (b) Graph $y_{1}=f(x)$ and $y_{2}=g(x)$ and use your result in part (a) to find the solution set of $f(x)<g(x)$ . Explain your answer. (c) Repeat part (b) for $f(x)>g(x)$

Find the $x$ - and $y$ -intercepts of the line whose standard form is $-2 x+5 y=-4$ . What is the slope of this line?

Suppose that an empty circular wading pool has a radius of 9 feet. During a storm, rain falling at a rate of 0.5 inch per hour begins to fill the pool. A small drain at the bottom of the pool is capable of draining 20 gallons of water per hour. (a) Determine the number of cubic inches of water falling into the pool in one hour. (Hint: Each hour a layer of water 0.5 inch thick falls into the pool.) (b) One gallon of water equals about 231 cubic inches. Write a formula for a function $g$ that computes the gallons of water landing in the pool in $x$ hours. (c) How many gallons of water land in the pool during a 2.5 hour storm? (d) Will the drain be able to keep up with the rainfall? If not, how many such drains would be needed?

Consider the linear function $f(x)=3(x-1)-\frac{1}{3}(6 x-9)$ . (a) Solve the equation $f(x)=0$ analytically. (b) Solve the inequality $f(x) \leq 0$ analytically. (c) Graph $y=f(x)$ in an appropriate viewing window and explain how the graph supports your answers in parts (a) and (b).

Consider the linear function $f(x)=\frac{1}{4}(5 x-8)-(x-3)$ . (a) Solve the equation $f(x)=0$ analytically. (b) Solve the inequality $f(x) \leq 0$ analytically. (c) Graph $y=f(x)$ in an appropriate viewing window and explain how the graph supports your answers in parts (a) and (b).

The graph below shows a line segment depicting the number of injuries and illnesses resulting in days away from work from 2004 to 2006. (Source: U.S. Department of Labor Statistics) (a) Use the midpoint formula to approximate the number of illnesses and injuries during the year 2005. (b) Find the slope of the line and explain its meaning in the context of this situation.

For each of the functions, determine the (i) domain (ii) range (iii) $x$ -intercept(s) (iv) $y$ -intercept(s). (a) (b) (c)

Use the screen to solve the equation or inequality. Here the function $y_{1}=f(x)$ is a linear function defined over the domain of real numbers. (a) $y_{1}=0$ (b) $y_{1}<0$ (c) $y_{1}>0$ (d) $y_{1} \leq 0$

Consider the linear functions $f(x)=\frac{3}{4}(4-x)+x-8$ and $g(x)=3(x-3)+2(x+2)$ . (a) Solve $f(x)=g(x)$ analytically, showing all steps. Also, check analytically. (b) Graph $y_{1}=f(x)$ and $y_{2}=g(x)$ and use your result in part (a) to find the solution set of $f(x)<g(x)$ . Explain your answer. (c) Repeat part (b) for $f(x)>g(x)$

Consider the linear functions $f(x)=-3(x+1)+2 x+1$ and $g(x)=2 x-2(3 x+10)$ . (a) Solve $f(x)=g(x)$ analytically, showing all steps. Also, check analytically. (b) Graph $y_{1}=f(x)$ and $y_{2}=g(x)$ and use your result in part (a) to find the solution set of $f(x)<g(x)$ . Explain your answer. (c) Repeat part (b) for $f(x)>g(x)$ .

Use the screen to solve the equation or inequality. Here the function $y_{1}=f(x)$ is a linear function defined over the domain of real numbers. (a) $y_{1}=0$ (b) $y_{1} \leq 0$ (c) $y_{1} \geq 0$ (d) $y_{1}>0$