Functions and Function Notation

1. A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.
2. When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.
3. When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.
4. Function
5. Function
6. Function
7. Function
8. [latex]f(−3) = -11[/latex]; [latex]f(2) = -1[/latex]; [latex]f(-a) = -2a-5[/latex]; [latex]-f(a) = -2a 5[/latex]; [latex]f(a h) = 2a 2h-5[/latex]
9. [latex]f(−3) = \sqrt{5} 5[/latex]; [latex]f(2) =2[/latex]; [latex]f(-a) = \sqrt{2 a} 5[/latex]; [latex]-f(a) = -\sqrt{2-a}-5[/latex]; [latex]f(a h) =\sqrt{2-a-h}-5[/latex]
10. [latex]f(−3) = 2[/latex]; [latex]f(2) = -2[/latex]; [latex]f(-a) = |-a−1|−|-a 1|[/latex]; [latex]-f(a) =-|a−1| |a 1|[/latex]; [latex]f(a h) =|a h−1|−|a h 1|[/latex]
11. Not a function
12. Function
13. Function
14. Not a function so it is also not a one-to-one function
15. One-to- one function
16. Function, but not one-to-one
17. Function
18. Function
19. Not a function
20. [latex]f(x)=1, x=2[/latex]
21. [latex]f(−2)=14; f(−1)=11; f(0)=8; f(1)=5; f(2)=2[/latex]
22. [latex]f(−2)=4; f(−1)=4.414; f(0)=4.732; f(1)=5; f(2)=5.236[/latex]
23. [latex]f(−2)=\frac{1}{9};f(−1)=\frac{1}{3};f(0)=1;f(1)=3;f(2)=9[/latex]
24. [latex]20[/latex]
25. [latex][0,100][/latex]

26. [latex][−0.001,0.001][/latex]

27. [latex][−1,000,000,1,000,000][/latex]

28. [latex][0,10][/latex]

29. [latex][−0.1,0.1][/latex]

30. [latex][−100,100][/latex]

31. 1. [latex]g(5000)=50[/latex]
    2. The number of cubic yards of dirt required for a garden of [latex]100[/latex] square feet is [latex]1[/latex].
32. 1. The height of a rocket above ground after [latex]1[/latex] second is [latex]200[/latex] ft.
    2. The height of a rocket above ground after [latex]2[/latex] seconds is [latex]350[/latex] ft.

Linear Functions

1. Terry starts at an elevation of [latex]3000[/latex] feet and descends [latex]70[/latex] feet per second.
2. [latex]d(t)=100−10t[/latex]
3. Yes
4. Yes
5. No
6. Increasing
7. Decreasing
8. Decreasing
9. [latex]2[/latex]
10. [latex]-2[/latex]
11. [latex]y=\frac{3}{5}x−1[/latex]
12. [latex]y=3x−2[/latex]
13. [latex]y=−\frac{1}{3}x \frac{11}{3}[/latex]
14. [latex]y[/latex]-int: [latex](0,2)[/latex]; [latex]x[/latex]-int: [latex](2,0)[/latex]
15. [latex]y[/latex]-int: [latex](0,-5)[/latex]; [latex]x[/latex]-int: [latex](\frac{5}{3},0)[/latex]
16. [latex]y[/latex]-int: [latex](0,4)[/latex]; [latex]x[/latex]-int: [latex](-10,0)[/latex]
17. Line 1: [latex]m = –10[/latex]; Line 2: [latex]m = –10[/latex]
18. Line 1: [latex]m = –2[/latex]; Line 2: [latex]m = 1[/latex]
19. Line 1: [latex]m=–2[/latex]; Line 2: [latex]m=–2[/latex]
20. [latex]y=3x−3[/latex]
21. [latex]y=−\frac{1}{3}t 2[/latex]
22. [latex]0[/latex]
23. [latex]y=−\frac{5}{4}x 5[/latex]
24. [latex]y=3x−1[/latex]
25. [latex]y=−2.5[/latex]
26. F
27. C
28. A
29. 
30. 
31. 

32. 
33. 
34. [latex]y=3[/latex]
35. [latex]x=−3[/latex]
36. Linear, [latex]g(x)=−3x 5[/latex]
37. Linear, [latex]f(x)=5x−5[/latex]
38. [latex]f(x)=−58x 17.3[/latex]
39. 
40. 1. [latex]a=11,900,b=1000.1[/latex]
    2. [latex]q(p)=1000p–100[/latex]
41. 
42. [latex]$45[/latex] per training session.
43. The rate of change is [latex]0.1[/latex]. For every additional minute talked, the monthly charge increases by [latex]$0.1[/latex] or [latex]10[/latex] cents. The initial value is [latex]24[/latex]. When there are no minutes used, initially the charge is [latex]$24[/latex].
44. The slope is [latex]–400[/latex]. this means for every year between 1960 and 1989, the population dropped by [latex]400[/latex] per year in the city.
45. C

Quadratic Functions

1. When written in that form, the vertex can be easily identified.
2. If [latex]a=0[/latex] then the function becomes a linear function.
3. If possible, we can use factoring. Otherwise, we can use the quadratic formula.
4. [latex]g(x)=(x 1)^2−4,[/latex]; Vertex: [latex](−1,−4)[/latex]
5. [latex]f(x)=(x 52)^2−\frac{33}{4}[/latex]; Vertex: [latex](−\frac{5}{2},−\frac{33}{4})[/latex]
6. [latex]f(x)=3(x−1)^2−12,[/latex]; Vertex: [latex](1,−12)[/latex]
7. Minimum is [latex]−\frac{17}{2}[/latex] and occurs at [latex]\frac{5}{2}[/latex]. Axis of symmetry is [latex]x=\frac{5}{2}[/latex].
8. Minimum is [latex]−\frac{17}{16}[/latex] and occurs at [latex]−\frac{1}{8}[/latex]. Axis of symmetry is [latex]x=−\frac{1}{8}[/latex].
9. Minimum is [latex]−\frac{7}{2}[/latex] and occurs at [latex]−3[/latex]. Axis of symmetry is [latex]x=−3[/latex].
10. Domain is [latex](−∞,∞)[/latex]. Range is [latex][2,∞)[/latex].
11. Domain is [latex](−∞,∞)[/latex]. Range is [latex][−5,∞)[/latex].
12. Domain is [latex](−∞,∞)[/latex]. Range is [latex][−12,∞)[/latex].
13. [latex]f(x)=x^2 4x 3[/latex]
14. [latex]f(x)=x^2−4x 7[/latex]
15. [latex]f(x)=−\frac{1}{49}x^2 \frac{6}{49}x \frac{89}{49}[/latex]
16. Vertex: [latex](3, −10)[/latex], axis of symmetry: [latex]x = 3[/latex], intercepts: [latex](3 +\sqrt{10},0)[/latex] and [latex](3-\sqrt{10},0)[/latex]

17. Vertex: [latex](\frac{7}{2},−\frac{37}{4})[/latex], axis of symmetry: [latex]x=\frac{7}{2}[/latex], [latex]y[/latex]-intercept: [latex](0,3)[/latex], [latex]x[/latex]-intercepts: [latex](\frac{7 \sqrt{37}}{2},0),(\frac{7-\sqrt{37}}{2},0)[/latex]

18. Vertex: [latex](\frac{3}{2},−12)[/latex], axis of symmetry: [latex]x=\frac{3}{2}[/latex], intercept: [latex]( \frac{3+2\sqrt{3}}{2},0)[/latex] and [latex]( \frac{3-2\sqrt{3}}{2},0)[/latex]
19. [latex]f(x)=x^2+2x+3[/latex]
20. [latex]f(x)=−3x^2−6x−1[/latex]
21. [latex]f(x)=x^2+2x+1[/latex]
22. [latex]f(x)=−x^2+2x[/latex]

Power Functions and Polynomial Functions

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
2. As [latex]x[/latex] decreases without bound, so does [latex]f(x)[/latex]. As [latex]x[/latex] increases without bound, so does [latex]f(x)[/latex].
3. The polynomial function is of even degree and leading coefficient is negative.
4. Power function
5. Neither
6. Neither
7. Degree = [latex]2[/latex], Coefficient = [latex]–2[/latex]
8. Degree = [latex]4[/latex], Coefficient =[latex]–2[/latex]
9. As [latex]x→∞[/latex], [latex]f(x)→∞[/latex], as [latex]x→−∞[/latex], [latex]f(x)→∞[/latex]
10. As [latex]x→−∞[/latex], [latex]f(x)→−∞[/latex], as [latex]x→∞[/latex], [latex]f(x)→−∞[/latex]
11. As [latex]x→−∞[/latex], [latex]f(x)→−∞[/latex], as [latex]x→∞[/latex], [latex]f(x)→−∞[/latex]
12. [latex]y[/latex]-intercept is [latex](0,12)[/latex], [latex]t[/latex]-intercepts are [latex](1,0)[/latex]; [latex](–2,0)[/latex]; and [latex](3,0)[/latex].
13. [latex]y[/latex]-intercept is [latex](0,−16)[/latex]. [latex]x[/latex]-intercepts are [latex](2,0)[/latex] and [latex](−2,0)[/latex].
14. [latex]y[/latex]-intercept is [latex](0,0)[/latex]. [latex]x[/latex]-intercepts are [latex](0,0)[/latex], [latex](4,0)[/latex], and [latex](−2, 0)[/latex].
15. 3
16. 5
17. 3
18. Yes. Number of turning points is [latex]2[/latex]. Least possible degree is [latex]3[/latex].
19. Yes. Number of turning points is [latex]1[/latex]. Least possible degree is [latex]2[/latex].
20. 
21. 
22. 
23. 
24. 

Graphs of Polynomial Functions

1. The [latex]x[/latex]-intercept is where the graph of the function crosses the [latex]x[/latex]-axis, and the zero of the function is the input value for which [latex]f(x)=0[/latex].
2. If we evaluate the function at [latex]a[/latex] and at [latex]b[/latex] and the sign of the function value changes, then we know a zero exists between [latex]a[/latex] and [latex]b[/latex].
3. There will be a factor raised to an even power.
4. [latex](−2,0), (3,0), (−5,0)[/latex]
5. [latex](3,0), (−1,0), (0,0)[/latex]
6. [latex](0,0), (−5,0), (2,0)[/latex]
7. [latex](0,0),(−5,0),(4,0)[/latex]
8. [latex]f(2)=–10[/latex] and [latex]f(4)=28[/latex]. Sign change confirms.
9. [latex]f(1)=3[/latex] and [latex]f(3)=–77[/latex]. Sign change confirms.
10. [latex]f(0.01)=1.000001[/latex] and [latex]f(0.1)=–7.999[/latex]. Sign change confirms.
11. [latex]0[/latex] with multiplicity [latex]2[/latex], [latex]−\frac{3}{2}[/latex] with multiplicity [latex]5[/latex], [latex]4[/latex] with multiplicity [latex]2[/latex]
12. [latex]0[/latex] with multiplicity [latex]2[/latex], [latex]–2[/latex] with multiplicity [latex]2[/latex]
13. [latex]−\frac{2}{3}[/latex] with multiplicity [latex]5[/latex], [latex]5[/latex] with multiplicity [latex]2[/latex]
14. [latex]x[/latex]-intercepts, [latex](1, 0)[/latex] with multiplicity [latex]2[/latex], [latex](–4, 0)[/latex] with multiplicity [latex]1[/latex], [latex]y[/latex]-intercept [latex](0, 4)[/latex]. As [latex]x→−∞[/latex], [latex]f(x)→−∞[/latex], as [latex]x→∞[/latex], [latex]f(x)→∞[/latex].

15. [latex]x[/latex]-intercepts [latex](3,0)[/latex] with multiplicity [latex]3[/latex], [latex](2,0)[/latex] with multiplicity [latex]2[/latex], [latex]y[/latex]-intercept [latex](0,–108)[/latex]. As [latex]x→−∞[/latex], [latex]f(x)→−∞[/latex], as [latex]x→∞[/latex], [latex]f(x)→∞[/latex].

16. [latex]x[/latex]-intercepts [latex](0, 0)[/latex], [latex](–2, 0),(4,0)[/latex] with multiplicity [latex]1[/latex], [latex]y[/latex]-intercept [latex](0, 0)[/latex]. As [latex]x→−∞[/latex], [latex]f(x)→∞[/latex], as [latex]x→∞[/latex], [latex]f(x)→−∞[/latex].

17. [latex]f(x)=−\frac{2}{9}(x−3)(x+1)(x+3)[/latex]
18. [latex]f(x)=\frac{1}{4}(x+2)^2(x−3)[/latex]
19. latex]–4, –2, 1, 3[/latex] with multiplicity [latex]1[/latex]
20. [latex]–2, 3[/latex] each with multiplicity [latex]2[/latex]
21. [latex]f(x)=−\frac{2}{3}(x+2)(x−1)(x−3)[/latex]
22. [latex]f(x)=\frac{1}{3}(x−3)^2(x−1)^2(x+3)[/latex]
23. [latex]f(x)=−\frac{1}{5}(x−1)^2(x−3)^3[/latex]
24. local max [latex](–.58, –.62)[/latex], local min [latex](.58, –1.38)[/latex]
25. global min [latex](–.63, –.47)[/latex]
26. global min [latex](.75, .89)[/latex]

Dividing Polynomials

1. [latex]x+6+\frac{5}{x−1}[/latex], quotient: [latex]x+6[/latex], remainder: [latex]5[/latex]
2. [latex]3x+2[/latex], quotient: [latex]3x+2[/latex], remainder: [latex]0[/latex]
3. [latex]x−5[/latex], quotient: [latex]x−5[/latex], remainder: [latex]0[/latex]
4. [latex]2x^2+2x+1+\frac{10}{x−4}[/latex]
5. [latex]2x^2−7x+1−\frac{2}{2x+1}[/latex]
6. [latex]3x^2−11x+34−\frac{106}{x+3}[/latex]
7. [latex]x2+5x+1[/latex]
8. Yes [latex](x−2)(3x^3−5)[/latex]
9. Yes [latex](x−2)(4x^3+8x^2+x+2)[/latex]
10. No
11. [latex](x−1)(x^2+2x+4)[/latex]
12. [latex](x−5)(x^2+x+1)[/latex]
13. Quotient: [latex]4x^2+8x+16[/latex], remainder: [latex]−1[/latex]
14. Quotient: [latex]3x2+3x+5[/latex], remainder: [latex]0[/latex]
15. Quotient: [latex]x^3−2x^2+4x−8[/latex], remainder: [latex]−6[/latex]

Zeros of Polynomial Functions

1. The theorem can be used to evaluate a polynomial.
2. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.
3. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.
4. [latex]-106[/latex]
5. [latex]0[/latex]
6. [latex]255[/latex]
7. [latex]-1[/latex]
8. [latex]−2, 1, 12[/latex]
9. [latex]−2[/latex]
10. [latex]-3[/latex]
11. [latex]−\frac{5}{2}, \sqrt{6}, −\sqrt{6}[/latex]
12. [latex]2, −4, −\frac{3}{2}[/latex]
13. [latex]4, −4, −5[/latex]
14. [latex]5, −3, −12[/latex]
15. [latex]\frac{1}{2}. \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}[/latex]
16. [latex]±5, ±1, ±\frac{5}{2}, ±\frac{1}{2}[/latex]
17. [latex]±1, ±\frac{1}{2}, ±\frac{1}{3}, ±\frac{1}{6}[/latex]
18. [latex]1, \frac{1}{2}, −\frac{1}{3}[/latex]
19. [latex]2, \frac{1}{4}, −\frac{3}{2}[/latex]