Pre-Calculus: Guided and Independent Practice: Piecewise-defined Functions

Question

PLEASE HELP! I have 3 more problems left on my Pre-Calculus homework and have no idea what to do. Please help me solve them! Thanks so much

1. f(x) = {2x + 5 if - 3 ≤ x < 0

{-3 if x = 0

{-5x if x > 0

a) Determine the domain of f.

b) Locate any intercepts

C) Graph f

D. Use the graph from part (c) to find the range of f.

2.. The function g is defined as:

g(x) = {1 + x, x < 0

{x^2, x ≥ 0

a) Determine the domain of g.

b) Locate intercepts

c. Graph g.

d. Use the graph from part (c) to find the range of g.

3.. The function g is defined as g(x) = { | x |, -2 ≤ x < 0

{x^3, x ≥ 0

a) Determine the domain of g.

b) Locate any intercepts

c. Graph g.

d. Use the graph from part (c) to find the range of g.

Answer 1

1. Draw the graph as follows: Piecewise functions are usually broken up. Expect it.

Put -3 into 2x + 5

2(-3) + 5 = -1 Put a solid dot on the graph at (-3, -1). It is a solid dot because there is a ≤ sign in the inequality between the -3 and the x.

Now put 0 into 2x + 5

2(0) + 5 = 5. Put an open dot at (0,5) on the graph. It is an open dot because there is NOT an equal sign between the x and the 0.

Connect the two dots with a straight line. Do not extend the line beyond either dot.

Next piece says that if x = 0, then the function (y) is -3. So put a solid dot on the point (0, -3)

The last part says that you may only draw this where x > 0. Note that it is a straight line with a slope of -5 and a y intercept of 0, So put an open dot at (0,0) then from that dot, count down 5 and to the right 1. That was so much fun, let's do it again. Count down 5 and to the right 1. Connect the dots but do NOT fill in the dot at (0,0). Extend the line to the right to complete your graph.

a. Determine the domain of f(x). That means describe all the values of x that HAVE a y value.

Look at your drawing from left to right. Your graph begins where x = -3 with a solid dot. Then all x's up to the axis have a y value on the line. At x = 0, there is a y value. It is not on the line but rather at -3. When x is larger than 0, every x value has a y value on the downward line. So your domain is [-3, ∞).

The -3 gets a square bracket because it IS included (thus the solid dot), then every x has a y from there to forever. Infinity ALWAYS gets a curved parenthesis because there is no stopping point. It just goes to infinity and beyond.

b. You are asked where this weird looking graph crosses the x and y axes. The upward line crosses the x axis at -2.5. So one x-intercept is at (-2.5, 0). Then the dot at (0, -3) is a y-intercept. The downward line never touches an axis and so does have an intercept. So the answers are (-2.5, 0) and (0, -3).

c. You have already drawn the graph.

d. The range is asking you for all the y values. Notice on the graph that the smallest y value is -∞ because the downward line goes on down forever. and the largest y value is almost 5. The actual number 5 cannot be included (round parenthesis) because the dot there is open. So the range is written in interval notation with the negative number first, then the positive. (-∞ , 5 ).

Whew! I went into great detail. I hope you could follow it. If not, please ask a question in the comment.

Will you still need help with numbers 2 and 3?