Determine whether the given function is linear

Math 3β€”College Algebra Homework 2.5β€”2.8 Name: ______________________ Show all your work for full credit: 2.5 Linear Functions and Models Q1β€”Q3. Determine whether the given function is linear. If the function is linear, express the function in the form 𝑓(π‘₯) = π‘Žπ‘₯ + 𝑏 Q1. 𝑓(π‘₯) = π‘₯(4 βˆ’ π‘₯) Q2. 𝑓(π‘₯) = π‘₯+1 5 Q3. 𝑓(π‘₯) = (π‘₯ + 1)2 Q4β€”Q5. For the given linear function, make a table of values and sketch its graph. What is the slope of the graph? Q4. 𝑓(π‘₯) = 2π‘₯ βˆ’ 5 2 Q5. π‘Ÿ(𝑑) = βˆ’ 𝑑 + 2 3 Q6β€”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(π‘₯) = 2π‘₯ βˆ’ 6 Q7. 𝑣(𝑑) = βˆ’ 10 3 𝑑 βˆ’ 20 Q8. The amount of trash in a country landfill is modeled by the function 𝑇(π‘₯) = 150π‘₯ + 32,000 Where π‘₯ is the number of years since 1996 and 𝑇(π‘₯) 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 Q1β€”Q4. Explain how the graph of g is obtained from the graph of 𝑓. Q1. 𝑓(π‘₯) = π‘₯ 2 𝑔(π‘₯) = (π‘₯ + 2)2 Q2. 𝑓(π‘₯) = π‘₯ 2 𝑔(π‘₯) = π‘₯ 2 + 2 Q3. 𝑓(π‘₯) = |π‘₯| 𝑔(π‘₯) = |π‘₯ + 2| βˆ’ 2 Q4. 𝑓(π‘₯) = |π‘₯| 𝑔(π‘₯) = |π‘₯ βˆ’ 2| + 2 Q5. Use the graph of 𝑦 = π‘₯ 2 to graph the follow. a) b) c) d) 𝑔(π‘₯) = π‘₯ 2 + 1 𝑔(π‘₯) = (π‘₯ βˆ’ 1)2 𝑔(π‘₯) = βˆ’π‘₯ 2 𝑔(π‘₯) = (π‘₯ βˆ’ 1)2 + 3 Q6β€”Q9. Sketch the graph of the function using transformations. Q6. 𝑓(π‘₯) = |π‘₯| βˆ’ 1 1 Q8. 𝑓(π‘₯) = 3 βˆ’ (π‘₯ βˆ’ 1)2 2 1 Q7. 𝑓(π‘₯) = 4 π‘₯ 2 1 Q9. 𝑓(π‘₯) = √π‘₯ + 4 βˆ’ 3 2 Q10β€”Q11. A function 𝑓 is given, write an equation for the final transformed graph. Q10. 𝑓(π‘₯) = |π‘₯|, shift 2 units to the left and shift downward 5 units. 4 Q11. 𝑓(π‘₯) = √π‘₯; reflect in the y-axis and shift upward 1 unit. Q12β€”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(π‘₯) = π‘₯ 4 Q13. 𝑓(π‘₯) = π‘₯ 2 + π‘₯ 2 2.7 Combining Functions Q1β€”Q4. Find 𝑓 + 𝑔, 𝑓 βˆ’ 𝑔, 𝑓𝑔, π‘Žπ‘›π‘‘ 𝑓/𝑔 and their domain Q1. 𝑓(π‘₯) = π‘₯ 2 + π‘₯, 𝑔(π‘₯) = π‘₯ 2 Q2. 𝑓(π‘₯) = 5 βˆ’ π‘₯, 𝑔(π‘₯) = π‘₯ 2 βˆ’ 3π‘₯ Q3. 𝑓(π‘₯) = √25 βˆ’ π‘₯ 2 , 𝑔(π‘₯) = √π‘₯ + 3 2 Q4. 𝑓(π‘₯) = π‘₯ , 4 𝑔(π‘₯) = π‘₯+4 Q5β€”Q7. Use 𝑓(π‘₯) = 2π‘₯ βˆ’ 3 and 𝑔(π‘₯) = 4 βˆ’ π‘₯ 2 to evaluate the expression. Q5. A) 𝑓(𝑔(0)) B) 𝑔(𝑓(0)) Q6. 𝐴) (𝑓°𝑔)(βˆ’2) B) (𝑔°𝑓)(βˆ’2) Q7. 𝐴) (𝑓°𝑔)(π‘₯) B) (𝑔°𝑓)(π‘₯) Q8—Q9. Find the functions 𝑓°𝑔, 𝑔°𝑓, 𝑓°𝑓, and 𝑔°𝑔 and their domains. Q8. 𝑓(π‘₯) = 1 π‘₯ Q9. 𝑓(π‘₯) = π‘₯ 2 𝑔(π‘₯) = 2π‘₯ + 4 𝑔(π‘₯) = π‘₯ + 1 3 2.8 One to One Functions and their Inverses Q1—Q3. Determine whether the function is one-to-one. Q1. 𝑓(π‘₯) = βˆ’2π‘₯ + 4 Q2. β„Ž(π‘₯) = π‘₯ 2 βˆ’ 2π‘₯ Q3. 𝑓(π‘₯) = √π‘₯ Q4—Q6. Assume that 𝑓 is a one-to-one function. Q4. if 𝑓(2) = 7, find 𝑓 βˆ’1 (7). Q5. 𝑖𝑓 𝑓 βˆ’1 (3) = βˆ’1, find 𝑓(βˆ’1) Q6. If 𝑓(π‘₯) = 5 βˆ’ 2π‘₯, find 𝑓 βˆ’1 (3) Q7β€”Q10. Use the inverse Function property to show that 𝑓 π‘Žπ‘›π‘‘ 𝑔 are inverse of each other. Q7. 𝑓(π‘₯) = π‘₯ βˆ’ 6 𝑔(π‘₯) = π‘₯ + 6 Q8. 𝑓(π‘₯) = 3π‘₯ + 4 𝑔(π‘₯) = Q9. 𝑓(π‘₯) = π‘₯ 2 βˆ’ 9, π‘₯ β‰₯ 0, π‘₯+2 Q10. 𝑓(π‘₯) = π‘₯βˆ’2 π‘₯βˆ’4 3 𝑔(π‘₯) = √π‘₯ + 9, π‘₯ β‰₯ βˆ’9 𝑔(π‘₯) = 2π‘₯+2 π‘₯βˆ’1 Q11β€”Q13 Find the inverse function of 𝑓. Q11. 𝑓(π‘₯) = 3π‘₯ + 5 Q12. 𝑓(π‘₯) = 2π‘₯+5 π‘₯βˆ’7 Q13. 𝑓(π‘₯) = 4 βˆ’ π‘₯ 2 , π‘₯ β‰₯ 0 Q14β€”Q15. A function 𝑓 is given. A) sketch the graph of 𝑓 B) use the graph of 𝑓 to sketch the graph of 𝑓 βˆ’1 C) Find 𝑓 βˆ’1 Q14. 𝑓(π‘₯) = 3π‘₯ βˆ’ 6 Q15. 𝑓(π‘₯) = √π‘₯ + 1 4
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