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Rational Functions
State the domain of the rational function.
f(x) = 7 14 - x A) (-∞, 14) ∪ (14, ∞) B) (-∞, - 7) ∪ (- 7, 7) ∪ (7, ∞) C) (-∞, 7) ∪ (7, ∞) D) (-∞, - 14) ∪ (- 14, 14) ∪ (14, ∞)
f(x) = x^ -^9 x2^ + 5 A) (-∞, 5) ∪ (5, ∞) B) (-∞, - 5) ∪ (-5, ∞) C) (-∞, - 3) ∪ (- 3, 3) ∪ (3, ∞) D) (-∞, ∞)
f(x) = (x^ -^ 3)(x^ +^ 4) x2^ - 1 A) (-∞, - 4) ∪ (-4, 3) ∪ (3, ∞) B) (-∞, - 1) ∪ (- 1, 1) ∪ (1, ∞) C) (-∞, ∞) D) (-∞, - 3) ∪ (-3, 4) ∪ (4, ∞)
f(x) = x^ -^5 x2^ - 1 A) (-∞, 1) ∪ (1, ∞) B) (-∞, - 1) ∪ (-1, ∞) C) (-∞, - 1) ∪ (- 1, 1) ∪ (1, ∞) D) (-∞, 5) ∪ (5, ∞)
f(x) = x^ -^9 x2^ + 7x A) (-∞, 9) ∪ (9, ∞) B) (-∞, - 7) ∪ (-7, ∞) C) (-∞, 7) ∪ (7, ∞) D) (-∞, - 7) ∪ (-7, 0) ∪ (0, ∞)
List the x- and y-intercepts, and graph the function.
f(x) = -^6 x - 6
f(x) = xx^ - +^23
f(x) = - 2x^ -^3 x + 2
-8 -4 4 8 x
y 8
4
-8 -4 4 8 x
y 8
4
PreCalculus
- f(x) = 3x^ -^5 x - 2
-8 -4 4 8 x
y 8
4
-8 -4 4 8 x
y 8
4
f(x) = x^ +^2 2x2^ - 3x - 2
f(x) = x^ -^1 x2^ - 5x - 6
f(x) = x
x + 2
f(x) = x x2^ - x - 6
f(x) = x
(^) - x - 6 x + 4
For the given function, find all asymptotes of the type indicated (if there are any)
- f(x) = x^ -^4 x2^ + 1
, vertical
A) x = 2, x = - 2 B) x = 1 C) None D) x = - 1
PreCalculus
- f(x) = x
(^) - x2 (^) + x - 1 x + 3
-10 -5 5 10 x
120 y 100 80 60 40 20
-10 -5 5 10 x
120 y 100 80 60 40 20
Answer Key
Testname: 2_RATIONAL_FUNCT
1) A
2) D
3) B
4) C
5) D
- No x-intercepts, y-intercept: 0, 1 ;
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
- x-intercept: 2, 0 ; y-intercept: 0, - 23 ;
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
- x-intercept: - 3 2
, 0 , y-intercept: 0, - 3 2
-8 -4 4 8 x
y 8
4
-8 -4 4 8 x
y 8
4
- x-intercept: 5 3
, 0 , y-intercept: 0, 5 2
-8 -4 4 8 x
y 8
4
-8 -4 4 8 x
y 8
4
- x-intercept: (-2, 0) , y-intercept: (0, - 1) ;
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
- x-intercept: (1, 0) , y-intercept: 0, 1 6
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
-10 -8 -6 -4 -2 2 4 6 8 10
10 8 6 4 2
Answer Key
Testname: 2_RATIONAL_FUNCT
-10 -5 5 10 x
y 5
-10 -5 5 10 x
y 5
x-intercept: (-1, 0) y-intercept: 0, - 1 5 Vertical asymptotes: x = - 5, x = 1 Horizontal asymptote: y = 0
lim x→- 5 -
f(x) = - ∞, lim x→- 5 +
f(x) = ∞, lim x→ 1 -
f(x) = - ∞,
lim x→ 1 +
f(x) = ∞
lim x→-∞
f(x) = 0, lim x→∞
f(x) = 0
Domain: (-∞, - 5) ∪ (-5, 1) ∪ (1, ∞) Range: (-∞, ∞) Continuity: all x ≠ - 5, 1 No local extrema Decreasing on: (-∞, - 5), (-5, 1), (1, ∞)
-10 -5 5 10 x
y 5
-10 -5 5 10 x
y 5
x-intercepts: (-2, 0) and (1, 0) y-intercept: 0, 1 2 Vertical asymptotes: x = - 1, x = 4 Horizontal asymptote: y = 1
lim x→- 1 -
f(x) = - ∞, lim x→- 1 +
f(x) = ∞, lim x→ 4 -
f(x) = - ∞,
lim x→ 4 +
f(x) = ∞
lim x→-∞
f(x) = 1, lim x→∞
f(x) = 1
Domain: (-∞, - 1) ∪ (-1, 4) ∪ (4, ∞) Range: (-∞, ∞) Continuity: all x ≠ - 1, 4 No local extrema Decreasing on: (-∞, - 1), (-1, 4), (4, ∞)
Answer Key
Testname: 2_RATIONAL_FUNCT
-10 -5 5 10 x
y 10
5
-10 -5 5 10 x
y 10
5
x-intercepts: (-2, 0) and (3, 0) y-intercept: (0, 3) Vertical asymptote: x = 2 Horizontal asymptote: none End-behavior asymptote: y = x + 1
lim x→ 2 -
f(x) = ∞, lim x→ 2 +
f(x) = - ∞
lim x→-∞
f(x) = - ∞, lim x→∞
f(x) = ∞
Domain: (-∞, 2) ∪ (2, ∞) Range: (-∞, ∞) Continuity: all x ≠ 2 No local extrema Increasing on: (-∞, 2), (2, ∞)
-10 -5 5 10 x
120 y 100 80 60 40 20
-10 -5 5 10 x
120 y 100 80 60 40 20
x-intercept: (1, 0) y-intercept: 0, - 1 3 Vertical asymptote: x = - 3 Horizontal asymptote: none End-behavior asymptote: y = x2^ - 4x + 13
lim x→- 3 -
f(x) = ∞, lim x→- 3 +
f(x) = - ∞
lim x→-∞
f(x) = ∞, lim x→∞
f(x) = ∞
Domain: (-∞, - 3) ∪ (-3, ∞) Range: (-∞, ∞) Continuity: all x ≠ - 3 Local minimum at (-4.7, 77) Increasing on: (-4.7, - 3), (-3, ∞) Decreasing on: (-∞, - 4.7)
