### MA 100: PrecalculusSolving Inequalities

Please work with a partner on this exercise. The purpose of this excercise is to review vertical shifts and reflections, solving quadratic equations, composition of functions and domains of square root functions while briefly covering section A.6 of the textbook.

1. Find a partner to work with. This project will be easier if you or your partner has a graphing calculator or computer. Follow the steps below to answer the question "What is the domain of the function f(x) = sqrt(x-1)?"
2. Graph the equation y = x - 1. Copy your graph below.

3. Darken the point where the line above crosses the x-axis. (This is the zero of the function g(x) = x-1.)
4. Darken the portion of the graph that lies above the x-axis. (These are the points on the graph of g(x) which have a positive y coordinate.)
5. Color the portion of the x-axis that lies below the darkened portions of the line.
6. The colored portion of the x-axis shows the values of x for which x-1 is zero or positive.
7. For what values of x is x - 1 greater than or equal to zero?

8. To check your work, write down the inequality "x - 1 is greater than or equal to 0", then add one to both sides. This is called solving the inequality.

9. At this point in the course the only valid inputs to the square root function are zero and positive numbers. Use what you know about the sign of x-1 to write the domain of the function f(x) = sqrt(x-1) below.

10. Graph the function f(x) = sqrt(x-1) to check your work. The graph of the function should lie above the portion of the x-axis that you colored in the graph above.
The next project is a quick review of equations of parabolas combined with some relatively questions about the domain of a square root. Taking the square root of the output of a function is an example of composition of functions, which we studied in section 1.7.
1. Use what you know about vertical shifts and reflections to graph the function h(x) = -x2 - 2 on the left below.

2. What are the zeros of h(x), if any?

3. For the parabola you just graphed, all points on the graph have negative y-coordinate. In other words, the output of h(x) is always negative. What is the domain of sqrt(h(x)) = sqrt(-x2 - 2)?

5. Write down a function f(x) whose graph is a parabola and whose output is always positive.

6. Graph your function f(x) on the right above.
7. What are the zeros of f(x), if any?

8. What is the domain of the function sqrt(f(x))? Check your answer by graphing.

In the final section of this worksheet, we look at quadratic functions that have both negative and positive output and consider composing the square root function with each of them.
1. Graph the function g(x) = x2 + 2x - 2 on the left below.

2. Use the graph to estimate the zeros of g(x). (How would you do this algebraically?) Darken the points on the graph where the zeros appear.

3. Darken all the points on the graph that have positive y-coordinates.
4. Color the points on the x-axis which lie below the darkened portion of the graph. These are the inputs for which the output of g(x) is positive.
5. Use the colored portions of the x-axis to write an (approximate) solution of the inequality "g(x) is greater than or equal to zero" below.

6. What is the domain of the function sqrt(g(x))?

7. Try to evaluate sqrt(g(-3)), sqrt(g(0)) and sqrt(g(1)). Which of -3, 0 and 5 are in the domain of sqrt(g(x))? Use this information to check your answer above.

At this point, you should know how to use the graph of a function f(x) to find the zeros of the function and which inputs give positive outputs. This allows you to estimate solutions of the inequality "f(x) is greater than or equal to 0".
1. Use a graph to estimate the solution of the equation -x2 - 2x + 2 < 0.

2. Use the quadratic formula to give an exact solution of the equation -x2 - 2x + 2 < 0.

3. What is the domain of sqrt(-x2 - 2x + 2)? (Be careful -- the answer to this question is not the same as the answer to the previous question.)

4. Check your answer using a graph or by plugging in values of x. Show your work below.

If you have time left over, try to find the equation of a function whose domain is:
1. All negative numbers.

2. A single point.

3. The interval from 0 to 1.

4. The interval from -2 to 1.

5. All real numbers except 5. (Hint: don't use a square root function.)

6. (Bonus) All negative numbers, 0, and the interval from 1 to 2.