Unit 1, Activity 4, Inverse Functions

Name______

Date______

1. Which of the functions graphed below have an inverse that is also a function? Explain why or why not using the horizontal line test.

2. Use a graphing utility to graph the functions given below and apply the horizontal line test to determine if the functions are one-to-one. Show a sketch of each graph. 3 a) y  1 b) y  x3  4x  6 x  2

c) y  x 2  5x d) y  23x

e) y  log x 2

Blackline Masters, Advanced Math – Pre-Calculus Page 30 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 4, Inverse Functions

3. Look at each of the tables given below. Decide which have an inverse that is also a function and give the numerical representation of that inverse function.

a) x -1 0 1 3 5 f(x 5 3 2 1 -3 )

b) x -3 -2 -1 0 1 f(x) 5 4 3 5 1

c) x -2 0 2 4 6 f(x -6 -4 -2 0 2 )

4. Functions f and g are inverse functions if f(g(x)) = g(f(x)) = x. Use this to determine whether or not the following are inverse functions. x  5 a) f (x)  3x  5 and g(x)  3 b) f (x)  3 x  2 and g(x)  x3  2 1 1 2x c) f (x)  and g(x)  x  2 x

Blackline Masters, Advanced Math – Pre-Calculus Page 31 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 4, Inverse Functions with Answers

1. Which of the functions graphed below have an inverse that is also a function? Explain why or why not using the horizontal line test.

a, d, f, g are one-to-one functions that will have an inverse. All are either strictly increasing or strictly decreasing functions and pass the horizontal line test. They are one-to-one functions.

2. Use a graphing utility to graph the functions given below and apply the horizontal line test to determine if the functions are one-to-one. 3 a) y  1 x  2 b) y  x3  4x  6 c) y  x 2  5x d) y  23x e) y  log x 2

Equations a and d have an inverse that is a function. Students should show the graph of each one and determine if they are one-to-one functions using the horizontal line test.

Blackline Masters, Advanced Math – Pre-Calculus Page 32 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 4, Inverse Functions with Answers

3. Look at each of the tables given below. Decide which have an inverse that is also a function and give the numerical representation of that inverse function. a) yes original inverse

x -1 0 1 3 5 x 5 3 2 1 -3 f(x 5 3 2 1 -3 f(x) -1 0 1 3 5 )

c) yes original inverse

x -2 0 2 4 6 x -6 -4 -2 0 2 f(x) -6 -4 -2 0 2 f(x -2 0 2 4 6 )

4. Functions f and g are inverse functions if f(g(x)) = g(f(x)) = x. Use this to determine whether or not the following are inverse functions. x  5 a) f (x)  3x  5 and g(x)  3 b) f (x)  3 x  2 and g(x)  x3  2 1 1 2x c) f (x)  and g(x)  x  2 x

All of the functions are inverses. Therefore, f(g(x)) = g(f(x)) = x.

Blackline Masters, Advanced Math – Pre-Calculus Page 33 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 5, Piecewise Defined Functions

Name______

Date______Part I. Given f(x). Graph each of the following. Is the graph continuous or discontinuous? How do you know? f (x)  1 for x  1 1) {x  2 for x  1

f (x)  x if x  -1 2) {x  3 if x > -1

f (x)  3 x for x  2 3) {2x 1 for x  2

Blackline Masters, Advanced Math – Pre-Calculus Page 34 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 5, Piecewise Defined Functions

Part II. Rewrite each of the absolute value functions as a piecewise defined function and then sketch the graph. Is the graph continuous or discontinuous? How do you know?

1) f(x) = |x+1|

2) g(x) = |2x - 4|

3) h(x) = |1 - x|

Blackline Masters, Advanced Math – Pre-Calculus Page 35 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 5, Piecewise Defined Functions with Answers

Part I. f (x)  1 for x  1 1) {x  2 for x  1 This is a discontinuous function. There is a hole in the graph at (1, -1) and a point at (1, 1).

f (x)  x if x  -1 2) {x  3 if x > -1 This is a discontinuous function. There is a jump at x = -1.

f (x)  3 x for x  2 3. {2x 1 for x  2 This is a discontinuous function. There is a jump at x = 2.

Blackline Masters, Advanced Math – Pre-Calculus Page 36 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 5, Piecewise Defined Functions with Answers

Part II

Blackline Masters, Advanced Math – Pre-Calculus Page 37 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, Activity 6, Library of Functions – Linear Functions

1. You are to create an entry for your Library of Functions. Introduce the parent function f(x) = x and include a table and graph of the function. Give a general description of the function written in paragraph form. Your description should include (a) the domain and range (b) local and global characteristics of the function – look at your glossary list and choose the words that best describe this function 2. Give some examples of family members using translation, reflection and dilation in the coordinate plane. Show these examples symbolically, graphically and numerically. Explain what has been done to the parent function to find each of the examples you give. 3. What are the common characteristics of a linear function? 4. Find a real-life example of how this function family can be used. Be sure to show a representative equation, table and graph. Does the domain and range differ from that of the parent function? If so, why? Describe what the slope, y-intercept, and zero mean within the context of your example. 5. Be sure that  your paragraph contains complete sentences with proper grammar and punctuation  your graphs are properly labeled, scaled, and drawn  you have used the correct math symbols and language in describing this function

Blackline Masters, Advanced Math – Pre-Calculus Page 38 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, General Assessments Spiral

1. Write an equation in standard form for

a) the line containing the points (1, 1) and (-1, 3).

4 b) a slope of and passing through the point (-1, 6). 3

2. Write an equation of a line parallel to 2x – 3y = 4 and passing through (1, 3).

3. Write an equation of a line perpendicular to x = 4 and passing through (4, 6).

4. Given f(x) = 3x2 + 2x – 4 find:

a) f(-1)

b) f(2x)

5. Solve the following system of equations: 3x + y = 13 2x – y = 2

6. Determine whether each of the following is a function. Explain.

a) {(-2, 3), (-1, 5), (3, 7), (4, 3)}

b) y2 = 4 – x2

7. Solve:

a) x2 – 8x – 9 = 0

b) 2x  4  4

c) |x – 6| = 8

Blackline Masters, Advanced Math – Pre-Calculus Page 39 Louisiana Comprehensive Curriculum, Revised 2008 Unit 1, General Assessments Spiral with Answers

1. Write an equation in standard form for a) the line containing the points (1, 1) and (-1, 3). x + y = 2 4 b) a slope of and passing through the point (-1, 6). 3 4x - 3y = -22

2. Write an equation of a line parallel to 2x – 3y = 4 and passing through (1, 3).

2x – 3y = -7 3. Write an equation of a line perpendicular to x = 4 and passing through (4, 6).

y = 6 4. Given f(x) = 3x2 + 2x – 4 find: a) f(-1) = -3 b) f(2x) = 12x2 +4x - 4

5. Solve the following system of equations: 3x + y = 13 2x – y = 2 (3, 4)

6. Determine whether each of the following is a function. Explain. a) {(-2, 3), (-1, 5), (3, 7), (4, 3)} Yes, each value of the domain {-2, -1, 3, 4} is used exactly one time b) y2 = 4 – x2 No, a distinct x, the domain, corresponds to two different values of y, the range. For example if x is 0, y is 2 or -2.

7. Solve: a) x2 – 8x – 9 = 0 solution: 9 or -1 b) 2x  4  4 solution: 6 c) |x – 6| = 8 solution: 14 or -2

Blackline Masters, Advanced Math – Pre-Calculus Page 40 Louisiana Comprehensive Curriculum, Revised 2008