Question

Please help, I'm really struggling to get my homework done

Answer 1

Domain is -1 and the range is 4.

Answer 2

1.

$f(x)=\dfrac1{x-1}=\dfrac1{x-1}+0$

x-1 = 0  for  x = 1

so:

Vertical asymptote: x = 1

Domain:  Dom f = R\{1}  {or: Dom f = (-∞, 1)∪(1, ∞)}

Horizontal asymptote:  y = 0

Range:    Ran f=R\{0}  {or: Ran f = (-∞, 0)∪(0, ∞)}

2.

$f(x)=\dfrac{-1}{x}+4=\dfrac{-1}{x-0}+4$

x-0 = 0  for  x = 0

so:

Vertical asymptote: x = 0

Domain:  Dom f = R\{0}  {or: Dom f = (-∞, 0)∪(0, ∞)}

Horizontal asymptote:  y = 4

Range:    Ran f=R\{4}  {or: Ran f = (-∞, 4)∪(4, ∞)}

3.

$f(x)=\dfrac{1}{x+3}-3$

x+3 = 0  for  x = -3

so:

Vertical asymptote: x = -3

Domain:  Dom f = R\{-3}  {or: Dom f = (-∞, -3)∪(-3, ∞)}

Horizontal asymptote:  y = -3

Range:    Ran f=R\{-3}  {or: Ran f = (-∞, -3)∪(-3, ∞)}

4.

$f(x)=\dfrac{-1}{x+5}-6$

x+5 = 0  for  x = -5

so:

Vertical asymptote: x = -5

Domain:  Dom f = R\{-5}  {or: Dom f = (-∞, -5)∪(-5, ∞)}

Horizontal asymptote:  y = -6

Range:    Ran f=R\{-6}  {or: Ran f = (-∞, -6)∪(-6, ∞)}

4.

$f(x)=\dfrac{-1}{x+1}+3$

x+1 = 0  for  x = -1

so:

Vertical asymptote: x = -1

Domain:  Dom f = R\{-1}  {or: Dom f = (-∞, -1)∪(-1, ∞)}

Horizontal asymptote:  y = -6

Range:    Ran f=R\{3}  {or: Ran f = (-∞, 3)∪(3, ∞)}

5.

$f(x)=\dfrac{1}{x+4}-2$

x+4 = 0  for  x = -4

so:

Vertical asymptote: x = -4

Domain:  Dom f = R\{-4}  {or: Dom f = (-∞, -4)∪(-4, ∞)}

Horizontal asymptote:  y = -2

Range:    Ran f=R\{-2}  {or: Ran f = (-∞, -2)∪(-2, ∞)}