Answer:
The domain = (-∞ , -1/4) ∪ (-1/4 , ∞)
The range = (-∞ , 21/4)∪(21/4 , ∞)
The answer is not in the choices
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 , ∞)