Solving 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.

- 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)*?" - Graph the equation
*y = x - 1*. Copy your graph below. - Darken the point where the line above crosses the
*x*-axis. (This is the zero of the function*g(x) = x-1*.) - 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.) - Color the portion of the
*x*-axis that lies below the darkened portions of the line. - The colored portion of
the
*x*-axis shows the values of*x*for which*x-1*is zero or positive. - For what values of
*x*is*x - 1*greater than or equal to zero? - 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**. - 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.

- 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.

- Use what you know about vertical shifts and reflections to graph the function
*h(x) = -x*on the left below.^{2}- 2 - What are the zeros of
*h(x)*, if any? - 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(-x*?^{2}- 2)

- Check your answer by graphing
*y = sqrt(-x^2 - 2)* - Write down a function
*f(x)*whose graph is a parabola and whose output is always positive.

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

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

- Graph the function
*g(x) = x*on the left below.^{2}+ 2x - 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. - Darken all the points on the graph that have positive
*y*-coordinates. - 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. - 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. - What is the domain of the function
*sqrt(g(x))*? - 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.

- Use a graph to estimate the solution of the equation
*-x*.^{2}- 2x + 2 < 0 - Use the quadratic formula to give an exact solution of the
equation
*-x*.^{2}- 2x + 2 < 0 - What is the domain of
*sqrt(-x*? (Be careful -- the answer to this question is not the same as the answer to the previous question.)^{2}- 2x + 2) - Check your answer using a graph or by plugging in values of
*x*. Show your work below.

- All negative numbers.
- A single point.
- The interval from 0 to 1.
- The interval from -2 to 1.
- All real numbers except 5. (Hint: don't use a square root function.)
- (Bonus) All negative numbers, 0, and the interval from 1 to 2.