# Determine whether the given function is linear

Math 3—College Algebra Homework 2.5—2.8 Name: ______________________ Show all your work for full credit: 2.5 Linear Functions and Models Q1—Q3. Determine whether the given function is linear. If the function is linear, express the function in the form 𝑓(𝑥) = 𝑎𝑥 + 𝑏 Q1. 𝑓(𝑥) = 𝑥(4 − 𝑥) Q2. 𝑓(𝑥) = 𝑥+1 5 Q3. 𝑓(𝑥) = (𝑥 + 1)2 Q4—Q5. For the given linear function, make a table of values and sketch its graph. What is the slope of the graph? Q4. 𝑓(𝑥) = 2𝑥 − 5 2 Q5. 𝑟(𝑡) = − 𝑡 + 2 3 Q6—Q7. A linear function is given. A) sketch the graph the rate of change of the function. b) Find the slope of the graph. C) Find Q6. 𝑓(𝑥) = 2𝑥 − 6 Q7. 𝑣(𝑡) = − 10 3 𝑡 − 20 Q8. The amount of trash in a country landfill is modeled by the function 𝑇(𝑥) = 150𝑥 + 32,000 Where 𝑥 is the number of years since 1996 and 𝑇(𝑥) is measured in thousands of tons. a) Sketch the graph of T b) What is the slope of the graph? c) At what rate is the amount of trash in the landfill increasing per year? 1 2.6 Transformations of functions Q1—Q4. Explain how the graph of g is obtained from the graph of 𝑓. Q1. 𝑓(𝑥) = 𝑥 2 𝑔(𝑥) = (𝑥 + 2)2 Q2. 𝑓(𝑥) = 𝑥 2 𝑔(𝑥) = 𝑥 2 + 2 Q3. 𝑓(𝑥) = |𝑥| 𝑔(𝑥) = |𝑥 + 2| − 2 Q4. 𝑓(𝑥) = |𝑥| 𝑔(𝑥) = |𝑥 − 2| + 2 Q5. Use the graph of 𝑦 = 𝑥 2 to graph the follow. a) b) c) d) 𝑔(𝑥) = 𝑥 2 + 1 𝑔(𝑥) = (𝑥 − 1)2 𝑔(𝑥) = −𝑥 2 𝑔(𝑥) = (𝑥 − 1)2 + 3 Q6—Q9. Sketch the graph of the function using transformations. Q6. 𝑓(𝑥) = |𝑥| − 1 1 Q8. 𝑓(𝑥) = 3 − (𝑥 − 1)2 2 1 Q7. 𝑓(𝑥) = 4 𝑥 2 1 Q9. 𝑓(𝑥) = √𝑥 + 4 − 3 2 Q10—Q11. A function 𝑓 is given, write an equation for the final transformed graph. Q10. 𝑓(𝑥) = |𝑥|, shift 2 units to the left and shift downward 5 units. 4 Q11. 𝑓(𝑥) = √𝑥; reflect in the y-axis and shift upward 1 unit. Q12—Q13. Determine whether the function 𝑓 is even, odd, or neither. If 𝑓 is even or odd, use symmetry to sketch the graph. Q12. 𝑓(𝑥) = 𝑥 4 Q13. 𝑓(𝑥) = 𝑥 2 + 𝑥 2 2.7 Combining Functions Q1—Q4. Find 𝑓 + 𝑔, 𝑓 − 𝑔, 𝑓𝑔, 𝑎𝑛𝑑 𝑓/𝑔 and their domain Q1. 𝑓(𝑥) = 𝑥 2 + 𝑥, 𝑔(𝑥) = 𝑥 2 Q2. 𝑓(𝑥) = 5 − 𝑥, 𝑔(𝑥) = 𝑥 2 − 3𝑥 Q3. 𝑓(𝑥) = √25 − 𝑥 2 , 𝑔(𝑥) = √𝑥 + 3 2 Q4. 𝑓(𝑥) = 𝑥 , 4 𝑔(𝑥) = 𝑥+4 Q5—Q7. Use 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 4 − 𝑥 2 to evaluate the expression. Q5. A) 𝑓(𝑔(0)) B) 𝑔(𝑓(0)) Q6. 𝐴) (𝑓°𝑔)(−2) B) (𝑔°𝑓)(−2) Q7. 𝐴) (𝑓°𝑔)(𝑥) B) (𝑔°𝑓)(𝑥) Q8—Q9. Find the functions 𝑓°𝑔, 𝑔°𝑓, 𝑓°𝑓, and 𝑔°𝑔 and their domains. Q8. 𝑓(𝑥) = 1 𝑥 Q9. 𝑓(𝑥) = 𝑥 2 𝑔(𝑥) = 2𝑥 + 4 𝑔(𝑥) = 𝑥 + 1 3 2.8 One to One Functions and their Inverses Q1—Q3. Determine whether the function is one-to-one. Q1. 𝑓(𝑥) = −2𝑥 + 4 Q2. ℎ(𝑥) = 𝑥 2 − 2𝑥 Q3. 𝑓(𝑥) = √𝑥 Q4—Q6. Assume that 𝑓 is a one-to-one function. Q4. if 𝑓(2) = 7, find 𝑓 −1 (7). Q5. 𝑖𝑓 𝑓 −1 (3) = −1, find 𝑓(−1) Q6. If 𝑓(𝑥) = 5 − 2𝑥, find 𝑓 −1 (3) Q7—Q10. Use the inverse Function property to show that 𝑓 𝑎𝑛𝑑 𝑔 are inverse of each other. Q7. 𝑓(𝑥) = 𝑥 − 6 𝑔(𝑥) = 𝑥 + 6 Q8. 𝑓(𝑥) = 3𝑥 + 4 𝑔(𝑥) = Q9. 𝑓(𝑥) = 𝑥 2 − 9, 𝑥 ≥ 0, 𝑥+2 Q10. 𝑓(𝑥) = 𝑥−2 𝑥−4 3 𝑔(𝑥) = √𝑥 + 9, 𝑥 ≥ −9 𝑔(𝑥) = 2𝑥+2 𝑥−1 Q11—Q13 Find the inverse function of 𝑓. Q11. 𝑓(𝑥) = 3𝑥 + 5 Q12. 𝑓(𝑥) = 2𝑥+5 𝑥−7 Q13. 𝑓(𝑥) = 4 − 𝑥 2 , 𝑥 ≥ 0 Q14—Q15. A function 𝑓 is given. A) sketch the graph of 𝑓 B) use the graph of 𝑓 to sketch the graph of 𝑓 −1 C) Find 𝑓 −1 Q14. 𝑓(𝑥) = 3𝑥 − 6 Q15. 𝑓(𝑥) = √𝑥 + 1 4
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