Piecewise, Square Root, Absolute Value
Problem Set 2: Other Functions (Piecewise, Square Root, Absolute Value) Directions: Complete the following problems on a separate sheet of graph or lined paper. You do not need to copy down the entire question, but make sure to write the number of each question. When you finish, mark all of your answers with a CHECK (correct) or CIRCLE (incorrect) and make any corrections needed on a separate, clean sheet. Key Terms:
Piecewise function Continuous Discontinuous Step function Absolute value function
Square root function Radicand Domain Range
I. Graph Square Root Functions 1. Given the function: 𝑓(𝑥) = √𝑥 a. Sketch a graph of 𝑓(𝑥). HINT: Use a table of values with the inputs 𝑥 = 0, 1, 4, 9, 16 if you get stuck. b. What is the domain of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑥′. DEFINITION: The DOMAIN of a function is its set of all possible 𝑥 values, e.g. Domain: {𝑥 ≥ 5} 2. For each function below: (a) identify the locator point, (b) sketch a graph of the function, (c) find the domain. If you get the hang of it, see if you can find the domain before sketching the graph. a. b. c. d. e.
𝑓(𝑥) = √𝑥 + 4 𝑔(𝑥) = √𝑥 − 3 ℎ(𝑥) = √𝑥 + 2 − 5 𝑚(𝑥) = −3√𝑥 𝑘(𝑥) = 3 − 2√𝑥 − 1
HINT: What transformations are taking place for each function? Recall the general form: 𝑦 = 𝑎√𝑥 − ℎ + 𝑘 WRITE: Which transformation (THINK: a, h, k) affects the domain? Why?
3. Another way to think of ‘domain’ is by looking at the equation of the function. Rather than looking at a graph, we can find the domain of a function algebraically (i.e. by solving an equation or inequality) by looking at the radicand. DEFINITION: The radicand is the number, term, or expression inside the square root. a. Which of the following values are defined? Undefined? √9
√0
√−3
√2
√−1
√𝑥
√2𝑥 − 5
b. [WRITE] What is the restriction we need to consider when taking the square root of a number? Specifically, what is the smallest possible radicand that will produce a defined value?
c. For what values of 𝑥 is the function 𝑓(𝑥) = √2𝑥 − 5 defined? In other words, what is the domain of 𝑓(𝑥)? HINT: What do you know about the value of 2𝑥 − 5 (the radicand)? What is the smallest value it can be? Write and solve an inequality using this fact. 4. Find the domain of each function below algebraically. a. b. c. d.
𝑓(𝑥) = √𝑥 − 3 𝑔(𝑥) = √2𝑥 + 8 ℎ(𝑥) = √3𝑥 − 5 𝑚(𝑥) = −3√4𝑥 + 3 + 1 HINT: Write an inequality to represent the possible values of the radicand.
II. Graph Absolute Value Functions 5. Given the function: 𝑓(𝑥) = |𝑥| a. Sketch a graph of 𝑓(𝑥). HINT: Use a table of values if you get stuck. b. What is the domain of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑥′. c. What is the range of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑦′. DEFINITION: The RANGE of a function is its set of all possible 𝑦 values, e.g. Range: {𝑦 ≥ 0} WRITE: What is the smallest possible output value of 𝑓(𝑥)? Why? For any absolute value function, what do you think the smallest possible output value is? 6. For each function below: (a) identify the vertex, (b) sketch a graph of the function, (c) find the range. If you get the hang of it, see if you can find the range before sketching the graph. a. b. c. d. e.
𝑓(𝑥) = |𝑥| − 2 𝑔(𝑥) = |𝑥 + 5| ℎ(𝑥) = |𝑥 − 3| + 1 𝑚(𝑥) = 4|𝑥| 𝑘(𝑥) = −2|𝑥 + 2| + 2
HINT: What transformations are taking place for each function? Recall the general form: 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 WRITE: Which transformation (THINK: a, h, k) affects the range? Why?
III. Graph Piecewise Functions 7. Use the piecewise function below to answer parts (a) and (b). 𝑓(𝑥) = {
𝑥 + 6, 𝑥2,
𝑥 < −2 −2 ≤ 𝑥 ≤ 2
a. Sketch a graph of the piecewise function above. b. What is the value of this function at 𝑥 = −2? What is the value of the function at 𝑥 = 2? c. Is the function continuous? Why or why not? 8. Write the equation of the piecewise function shown below. Make sure to include the domain of each function.
HINT: How many separate functions are contained in this piecewise function? What types of functions are they? What are the key points of each function? What are the domains and ranges of each function? 9. Below is the graph of a function, 𝑓(𝑥). Answer all parts below.
a. b. c. d. e. f.
What is 𝑓(−1)? What is 𝑓(0)? What is 𝑓(1)? What is 𝑓(4)? What is 𝑓(0.5)? What is the domain of 𝑓(𝑥)?
IV. Identify and Graph Step Functions Question 10 relies on the graph below.
10. The total cost (in dollars) of a taxi ride as a function of the length of the trip (in miles) is represented by the graph above. For each trip length below, determine the approximate cost (in dollars) of the ride. a. b. c. d. e.
3.5 miles 4 miles 0 miles 6.5 miles 8 miles
Piecewise function Continuous Discontinuous Step function Absolute value function
Square root function Radicand Domain Range
I. Graph Square Root Functions 1. Given the function: 𝑓(𝑥) = √𝑥 a. Sketch a graph of 𝑓(𝑥). HINT: Use a table of values with the inputs 𝑥 = 0, 1, 4, 9, 16 if you get stuck. b. What is the domain of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑥′. DEFINITION: The DOMAIN of a function is its set of all possible 𝑥 values, e.g. Domain: {𝑥 ≥ 5} 2. For each function below: (a) identify the locator point, (b) sketch a graph of the function, (c) find the domain. If you get the hang of it, see if you can find the domain before sketching the graph. a. b. c. d. e.
𝑓(𝑥) = √𝑥 + 4 𝑔(𝑥) = √𝑥 − 3 ℎ(𝑥) = √𝑥 + 2 − 5 𝑚(𝑥) = −3√𝑥 𝑘(𝑥) = 3 − 2√𝑥 − 1
HINT: What transformations are taking place for each function? Recall the general form: 𝑦 = 𝑎√𝑥 − ℎ + 𝑘 WRITE: Which transformation (THINK: a, h, k) affects the domain? Why?
3. Another way to think of ‘domain’ is by looking at the equation of the function. Rather than looking at a graph, we can find the domain of a function algebraically (i.e. by solving an equation or inequality) by looking at the radicand. DEFINITION: The radicand is the number, term, or expression inside the square root. a. Which of the following values are defined? Undefined? √9
√0
√−3
√2
√−1
√𝑥
√2𝑥 − 5
b. [WRITE] What is the restriction we need to consider when taking the square root of a number? Specifically, what is the smallest possible radicand that will produce a defined value?
c. For what values of 𝑥 is the function 𝑓(𝑥) = √2𝑥 − 5 defined? In other words, what is the domain of 𝑓(𝑥)? HINT: What do you know about the value of 2𝑥 − 5 (the radicand)? What is the smallest value it can be? Write and solve an inequality using this fact. 4. Find the domain of each function below algebraically. a. b. c. d.
𝑓(𝑥) = √𝑥 − 3 𝑔(𝑥) = √2𝑥 + 8 ℎ(𝑥) = √3𝑥 − 5 𝑚(𝑥) = −3√4𝑥 + 3 + 1 HINT: Write an inequality to represent the possible values of the radicand.
II. Graph Absolute Value Functions 5. Given the function: 𝑓(𝑥) = |𝑥| a. Sketch a graph of 𝑓(𝑥). HINT: Use a table of values if you get stuck. b. What is the domain of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑥′. c. What is the range of 𝑓(𝑥)? Write your answer as an inequality (< 𝑜𝑟 >) containing ′𝑦′. DEFINITION: The RANGE of a function is its set of all possible 𝑦 values, e.g. Range: {𝑦 ≥ 0} WRITE: What is the smallest possible output value of 𝑓(𝑥)? Why? For any absolute value function, what do you think the smallest possible output value is? 6. For each function below: (a) identify the vertex, (b) sketch a graph of the function, (c) find the range. If you get the hang of it, see if you can find the range before sketching the graph. a. b. c. d. e.
𝑓(𝑥) = |𝑥| − 2 𝑔(𝑥) = |𝑥 + 5| ℎ(𝑥) = |𝑥 − 3| + 1 𝑚(𝑥) = 4|𝑥| 𝑘(𝑥) = −2|𝑥 + 2| + 2
HINT: What transformations are taking place for each function? Recall the general form: 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 WRITE: Which transformation (THINK: a, h, k) affects the range? Why?
III. Graph Piecewise Functions 7. Use the piecewise function below to answer parts (a) and (b). 𝑓(𝑥) = {
𝑥 + 6, 𝑥2,
𝑥 < −2 −2 ≤ 𝑥 ≤ 2
a. Sketch a graph of the piecewise function above. b. What is the value of this function at 𝑥 = −2? What is the value of the function at 𝑥 = 2? c. Is the function continuous? Why or why not? 8. Write the equation of the piecewise function shown below. Make sure to include the domain of each function.
HINT: How many separate functions are contained in this piecewise function? What types of functions are they? What are the key points of each function? What are the domains and ranges of each function? 9. Below is the graph of a function, 𝑓(𝑥). Answer all parts below.
a. b. c. d. e. f.
What is 𝑓(−1)? What is 𝑓(0)? What is 𝑓(1)? What is 𝑓(4)? What is 𝑓(0.5)? What is the domain of 𝑓(𝑥)?
IV. Identify and Graph Step Functions Question 10 relies on the graph below.
10. The total cost (in dollars) of a taxi ride as a function of the length of the trip (in miles) is represented by the graph above. For each trip length below, determine the approximate cost (in dollars) of the ride. a. b. c. d. e.
3.5 miles 4 miles 0 miles 6.5 miles 8 miles
