Find the domain and range ! 10 points math 3

The domain = (-∞ , -1/4) ∪ (-1/4 , ∞)

The range = (-∞ , 21/4)∪(21/4 , ∞)

The answer is not in the choices

Step-by-step explanation:

* Lets revise how to find the inverse function

- At first write the function as y = f(x)

- Then switch x and y

- Then solve for y

- The domain of f(x) will be the range of f^-1(x)

- The range of f(x) will be the domain of f^-1(x)

* Now lets solve the problem

- At first find the domain and the range of f(x)

∵ f(x) = (x - 9)/(21 - 4x)

- The domain is all real numbers except the value which

  makes the denominator = 0

- To find this value put the denominator = 0

∴ 21 - 4x = 0 ⇒ subtract 21 from both sides

∴ -4x = -21 ⇒ ÷ -4 both sides

∴ x = 21/4

∴ The domain = R - {21/4} OR the domain = (-∞ , 21/4)∪(21/4 , ∞)

* Now lets find the range

- The range will be all the values of real numbers except -1/4

 because the horizontal asymptote equation is y = -1/4

- To find the horizontal asymptote we find the equation y = a/b

 where a is the coefficient of x up and b is the coefficient of x down

∵ The coefficient of x up is 1 and down is -4

∴ The equation y = 1/-4

∴ The value of y = -1/4 does not exist

∴ The range = R - {-1/4} OR the range = (-∞ , -1/4) ∪ (-1/4 , ∞)

* Switch the domain and the range for the f^-1(x)

∴ The domain = (-∞ , -1/4) ∪ (-1/4 , ∞)

∴ The range = (-∞ , 21/4)∪(21/4 , ∞)

d. all of the above

step-by-step explanation: