This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Precalculus exercises

## Algebra etc.

• Solve the quadratic equation $x^{2}+2x+10=0$ and represent the answer in the standard form $a+bi$, where $a$ and $b$ are real numbers.
• Solve the following equation: $\sqrt{x^2-7}-3=0$.
• Solve the following equation: $\left( x^2-7 \right)^3 -8=0$.
• Consider the parabola below. Find its vertex, its axis of symmetry, and its maximum or minimum.
• Find the exact values of the $x$-coordinates of the intersections between the parabola and the line below:
• There are $125$ sheep and $5$ dogs in a flock. How old is the shepherd?
• Let $h(x)=x^2+3x-10$. Find the $x$- and $y$-intercepts and sketch the graph of the function.
• Find an equation the solutions of which are $1$ and $2$.

## Coordinate system

• Find the equation of the line passing through the points $(-1,1)$ and $(-1,5)$.
• What is the distance from the center of the circle $(x-1)^2+(y+3)^2=5$ to the origin?
• What is the distance from the the circle $x^2+(y+3)^2=2$ to the origin?
• Find the equation of the circle centered at $(-1,-1)$ and passing through the point $(-1,1)$.
• Three straight lines are shown below. Find their slopes:
• Three straight lines are shown below. Find their equations:

• Set up, but do not solve, a system of linear equations for the following problem: “Suppose your portfolio is worth $\$ 1,000,000$and it consists of two stocks$A$and$B$. The stocks are priced as follows:$A\$2.1$ per share, $B$ $\$1.5$per share. Suppose also that you have twice as much of stock$A$than$B$. How much of each do you have?” • In an effort to find the point in which the lines$ 2x -y=2 $and$-4x+2y=1$intersect, a student multiplied the first one by$2$and then added the result to the second. He got$0=5$. Explain the result. • Vectors$A$and$B$are given below. Copy the picture and illustrate graphically: (a)$A+B$, (b)$A-B$, (c)$||A||$. • Find the angle between the vectors$<1,1>$and$<1,2>$. Don't simplify. • Solve the system of linear equations: $$\left\{\begin{array}{lll} x&-y&=-1,\\ 2x&+y&=0.\\ \end{array}\right.$$ ## Polynomials • Find the line passing through the point$(-1,2)$and perpendicular to the line$y=-x-2016.$• Suppose$f$is a polynomial of degree$55$and its leading term is$-1$. Describe the long term behavior of this function. • For the polynomial$f(x)=-2x^2(x+2)^2(x^2+1)$, find its$x$-intercepts. • Find a formula for a polynomial with these roots:$1$,$2$, and$3$. • Find the equation satisfied by all points that lie$2$units away from the point$(-1,-2)$and by no other points. • For the polynomials graphed below, find the following: $$\begin{array}{r|lll} &1&2&3\\ \hline \text{the smallest possible degree }&\\ \text{the sign of the leading coefficient }&\\ \text{the degree is odd/even } \end{array}$$ • Find a possible formula for the function plotted below: • Find a possible formula for the function plotted below: • For the polynomial$f(x)=-2x(x-2)^2(x+1)^3$, find its$x$-intercepts and its large scale behavior, i.e.,$f(x)\to ?$as$x\to \pm \infty$. • Given$f(x)=-(x-3)^{4}(x+1)^{3}$. Find the leading term and use it to describe the long term behavior of the function. • (a) Solve the equation$(x^2+1)(x+1)(x-1)=0$. (b) Solve the inequality$(x^2+1)(x+1)(x-1)>0$. ## Functions • The perimeter of a rectangle is$10$feet. (a) Express the area of the rectangle in terms of its width. (b) Find the minimal possible area. (c) Find the maximal possible area. • The area of a rectangle is$100$sq. feet. (a) Express the perimeter of the rectangle in terms of its width. (b) Find the minimal possible perimeter. (c) Find the maximal possible perimeter. • The graph of the function$y=f(x)$is given below. (a) Find such a$y$that the point$(2,y)$belongs to the graph. (b) Find such an$x$that the point$(x,3)$belongs to the graph. (b) Find such an$x$that the point$(x,x)$belongs to the graph. Show your drawing. • Make a hand-drawn sketch of the graph of the function: $$f(x)= \begin{cases} -3 &\text{ if } x<0\\ x^2 &\text{ if } 0\le x<1\\ x &\text{ if } x>1 \end{cases}$$ • Find the implied domains of the functions given by: $$\text{(a) } \frac{x+1}{\sqrt{x^2-1}};\ \text{(b) }\sqrt[4]{x+1}.$$ • Find the implied domain of the function given by: $$\frac{1}{(x-1)(x^2+1)}.$$ • Find the implied domain of the function given by: $$\frac{1}{\sqrt{x+1}}.$$ • Find the implied domain of the function given by: $$(x-1)(x^2+1)2^x.$$ • Finish the sentence: “If a function fails the horizontal line test, then...” • Restate (but do not solve) the following problem algebraically: “What are the dimensions of the rectangle with the smallest possible perimeter and area fixed at$100$?” • A sketch of the graph of a function$f$and its table of values are given below. Complete the table: $$\begin{array}{r|ll} x&0& &3& &1\\ \hline y&2&4& &5& \end{array}$$ • Plot the graph of the function$y=f(x)$, where$x$is the income (in thousands of dollars) and$f(x)$is the tax bill (in thousands of dollars) for the income of$x$, which is computed as follows: no tax on the first$\$10,000$, then $5\%$ for the next $\$ 10,000$, and$10\%$for the rest of the income. • Plot the graph of the function$y=f(x)$, where$x$is time in hours and$y=f(x)$is the parking fee over$x$hours, which is computed as follows: free for the first hour, then$\$1$ per every full hour for the next $3$ hours, and a flat fee of $\$5$for anything longer. • Explain the difference between these two functions: $$\sqrt{\frac{x-1}{x+1}}\text{ and } \frac{\sqrt{x-1}}{\sqrt{x+1}}.$$ ## Features of graphs • Is the function below even, odd, or neither? $$f(x)=\frac{x}{e^x-1}+\frac{1}{2}x-1$$ • Give an example of an even function, an odd function, and a function that's neither. Provide formulas. • Test whether the following three functions are even, odd, or nether: (a)$f(x)=x^3+1$, (b) the function the graph of which is a parabola shifted one unit up, (c) the function with this graph: • Find horizontal asymptotes of these functions: • Is$\sin x/2$a periodic function? If it is, find its period. You have to justify your conclusion algebraically. • Is$\sin x+\cos \pi x$a periodic function? If it is, find its period. You have to justify your conclusion algebraically. • Is$\sin x+\sin 2x$a periodic function? If it is, find its period. You have to justify your conclusion algebraically. • (a) State the definition of a periodic function. (b) Give an example of a periodic polynomial. • Prove, from the definition, that the function$f(x)=x^2+1$is increasing for$x>0$. • The graph of a function$f(x)$is given below. (a) Find$f(-4)$,$f(0)$, and$f(4)$. (b) Find such an$x$that$f(x)=2$. (c) Is the function one-to-one? • A sketch of the graph of a function$f$is given below. Describe its behavior the function using words “decreasing” and “increasing”. • The graph of the function$y=f(x)$is given below. (1) Find its domain. (2) Determine intervals on which the function is decreasing or increasing. (3) Provide$x$-coordinates of its relative maxima and minima. (4) Find its asymptotes. ## Compositions • Find the composition$h(x)=(g\circ f)(x)$of the functions$y=f(x)=x^{2}-1$and$g(y)=3y-1.$Evaluate$h(1)$. • Represent the function$h(x)=2\sin^3x+\sin x+5$as the composition of two functions one of which is trigonometric. • (a) Represent function$h(x)=e^{x^3-1}$, as the composition of two functions$f$and$g$, (b) Provide formulas for the two possible compositions of the two functions: “take the logarithm of” and “take the square root of”. • Suppose function$f$performs the operation: “take the logarithm of”, and function$g$performs: “take the square root of”. (a) Verbally describe the inverses of$f$and$g$. (b) Find the formulas for these four functions. (c) Find their domains. • (a) Represent function$h(x)=\sqrt{x^2-1}$as the composition of two functions$f$and$g$. (b) Provide a formula for the composition$y=f(g(x))$of$f(u)=u^{2}+u$and$g(x)=2x-1$. • Provide a formula for the composition$y=f(g(x))$of$f(u)=\sin u$and$g(x)=\sqrt{x}$. • Provide a formula for the composition$y=f(g(x))$of$f(u)=u^{2}-3u+2$and$g(x)=x$. • Find the inverse of the function$f(x)= 3x^2+1$. Choose appropriate domains for these two functions. • (a) Represent the function$h(x)=\sqrt{x-1}$as the composition of two functions. (b) Represent the function$k(t)=\sqrt{t^2-1}$as the composition of three functions. (c) Represent the function$p(t)=\sin\sqrt{t^2-1}$as the composition of four functions. • (a) What is the composition$f\circ g$for the functions given by$f(u)=u^2+u$and$g(x)=3$? (a) What is the composition$f\circ g$for the functions given by$f(u)=2$and$g(x)=\sqrt{x}$? • Is the composition of two functions that are odd/even odd/even? • Represent this function:$h(x)=\frac{x^3+1}{x^3-1},$as the composition of two functions of variables$x$and$y$. • Represent the composition of these two functions:$f(x)=\frac{1}{x}+1$and$g(y)=\sqrt{y-1}$, as a single function$h$of variable$x$. Don't simplify. • Function$y=f(x)$is given below by a list its values. Find its inverse and represent it by a similar table. $$\begin{array}{r|l|l|l}x &0 &1 &2 &3 &4 \\\hline y=f(x) &1 &2 &0 &4 &3 \end{array}$$ • What is the function that is its own inverse? • Plot the inverse of the function shown below, if possible. • Plot the graph of the inverse of this function: • Represent this function:$h(x)=\tan (2x)$as the composition of two functions of variables$x$and$y$. • Find the composition$h(x)=(g\circ f)(x)$of the functions$y=f(x)=x^{2}-1$and$g(y)=\frac{y-1}{y+1}$. Evaluate$h(0)$. • Functions$y=f(x)$and$u=g(y)$are given below by tables of some of their values. Present the composition$u=h(x)$of these functions by a similar table: $$\begin{array}{r|c|c|c|c} x &0 &1 &2 &3 &4 \\ \hline y=f(x) &1 &1 &2 &0 &2 \end{array}$$ $$\begin{array}{c|c|c|c|c} y &0 &1 &2 &3 &4 \\ \hline u=g(y) &3 &1 &2 &1 &0 \end{array}$$ • Find the composition$h(x)=(g\circ f)(x)$of the functions$y=f(x)=x^{2}-1$and$g(y)=3y-1.$Evaluate$h(1)$. • Represent the composition of these two functions:$f(x)=1/x$and$g(y)=\frac{y}{y^2-3}$, as a single function$h$of variable$x$. Don't simplify. • Represent this function:$h(x)=\frac{x^3+1}{x^3-1},$as the composition of two functions of variables$x$and$y$. • Function$y=f(x)$is given below by a list of its values. Is the function one-to one? What about its inverse? $$\begin{array}{r|l|l|l|l|l}x&0 &1 &2 &3 &4 \\ \hline y=f(x)&0 &1 &2 &1 &2 \end{array}$$ • Function$y=f(x)$is given below by a list of its values. Is the function one-to one? What about its inverse? $$\begin{array}{r|l|l|l|l|l}x&0 &1 &2 &3 &4 \\ \hline y=f(x)&7 &5 &3 &4 &6 \end{array}$$ • Function$y=f(x)$is given below by a list of some of its values. Add missing values in such a way that the function is one-to one. $$\begin{array}{r|l|l|l|l|l}x&-1 &0 &1 &2 &3 &4 &5\\ \hline y=f(x)&-1 & &4 &5 & &2 \end{array}$$ • Plot the graph of the function$f(x)=\frac{1}{x-1}$and the graph of its inverse. Identify its important features. • (a) Algebraically, show that the function$f(x)=x^{2}$is not one-to-one. (b) Graphically, show that the function$g(x)=2^{x+1}$is one-to-one. (c) Find the inverse of$g$. • Find the formulas of the inverses of the following functions: (a)$f(x)=(x+1)^3$, (b)$g(x)=\ln (x^3)$. • Sketch the graph of the inverse of the function below: ## Transformations of functions • The graph drawn with a solid line is$y=x^3$. What are the other two? • The graph of one the functions below is$y=e^x$. What is the other? • The graphs below are parabolas. One is$y=x^2$. What is the other? • The graph below is the graph of the function$f(x)=A\sin x+B$for some$A$and$B$. Find these numbers. • The graph of function$f$is given below. Sketch the graph of$y=2f(x+2)+2$. Explain how you get it. • By transforming the graph of$y=e^x$, plot the graph of the function$f(x)=2e^{x-3}$. Identify the domain, the range, and the asymptotes. • One of the graphs below is that of$y=\arctan x$. What are the others? • Half of the graph of an even function is shown below; provide the other half: • Half of the graph of an odd function is shown below; provide the other half: • The graph of the function$y=f(x)$is given below. Sketch the graph of$y=2f(x)$and then$y=2f(x)-1$. • What is the relation between these two functions? • Plot the graph of a function that is both odd and even. • Give examples of odd and even functions that aren't polynomials. • Is the inverse of an odd/even function odd/even? • By transforming the graph of$y=\sin x$, plot the graph of the function$f(x)=2\sin (x-3)$. Identify the domain, the range, and the asymptotes. • Give examples of an even function, an odd function, and a function that's neither. Provide formulas. • The graph below is a parabola. Find its formula: ## Early models • The population of a city has doubled in$10$years. Assuming exponential growth, how long does it take to triple? • The population of a city has doubled in$10$years. Assuming exponential growth, how much does it grow every year? • Provide a formula for modeling radioactive decay. What is the half-life of an element? • The population of a city declines by$10\%$every year. How long will it take to drop to$50\%$of the current population? • The function$y=f(x)$shown below represents the location (in miles) of a hiker as a function of time (in hours). Find the hiker's average velocity. • A city loses$3\%$of its population every year. How long will it take to lose$20\%$? • Find the average rate of change of the function shown below on the interval$[1,6]$: • A car start moving east from town A at a constant speed of$60$miles an hour. Town B is located$10$miles south of A. Represent the distance from town B to the car as a function of time. ## Exponentials etc. • Solve the equation:$2^x=3^{x+1}$. Don't simplify. • Solve the equation:$3^x=3^{x+1}$. • Solve the equation:$3^x=2$. • Solve the equation:$2^x=2\cdot 3^{x+1}$. Don't simplify. • To what power should you raise$3$to get$10$? • Find the domain, the range, and the asymptotes of the function$f(x)=\ln (x-3)+\ln 3$. ## Other • Given vectors$a=<1,2>,\ b=<-2,1>$, find their magnitudes and the angle between them. • Compute$\displaystyle\sum _{n=1}^{4} n^2$. • Find the equation of the line starting at the point$(1,2,3)$in the direction of the vector$<1,1,1>$. • Set up a system of linear equations -- but do not solve it -- for the following problem: “An investment portfolio worth$\$1,000,000$ is to be formed from the shares of: Microsoft - $\$5$per share and Apple -$\$7$ per share. If you need to have twice as many shares of Microsoft than Apple, what are the numbers?”
• Set up a system of linear equations -- but do not solve -- for the following problem: “A mix of coffee is to be prepared from: Kenyan coffee - $\$3$per pound and Colombian coffee -$\$5$ per pound. How much of each do you need to have $10$ pounds of blend with $\$3.50$per pound?” • Set up, do not solve, the system of linear equations for the following problem: “One serving of tomato soup contains$100$Cal and$18$g of carbohydrates. One slice of whole bread contains$70$Cal and$13$g of carbohydrates. How many servings of each should be required to obtain$230$Cal and$42\$ g of carbohydrates?”
• Solve the system of linear equations:

$$\begin{cases} x-y&=2,\\ x+2y&=1. \end{cases}$$

• Solve the system of linear equations and geometrically represent its solution:

$$\begin{cases} x-2y&=1,\\ x+2y&=-1. \end{cases}$$

• Geometrically represent this system of linear equations:

$$\begin{cases} x-2y&=1,\\ x+2y&=-1. \end{cases}$$

• What are the possible outcomes of a system of linear equations?