MATH 3 San Diego State University Linear Functions and Models Questions

Math 3—College AlgebraHomework 2.5—2.8
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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
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