Find the domain and range for the inverse of f(x) = (1/4x)+2.

Ha! Finding the domain and range for the inverse function, huh? Well, let's clown around with it!

To find the domain and range of the inverse of f(x) = (1/4x) + 2, we first need to find the inverse function itself. So, let's switch things up and solve for x instead of y.

Start by swapping x and y:
x = (1/4y) + 2

Now, let's solve for y:
4x - 8 = y

So, the inverse function is: f^(-1)(x) = 4x - 8

Now, to determine the domain and range, let's play with these!

The domain of f(x) is all real numbers except 0, since dividing by zero is a big no-no. Therefore, the domain of f^(-1)(x) is also all real numbers except 0.

As for the range, since f(x) is a linear function, it covers the entire real number line. Therefore, the range of f^(-1)(x) is also the entire real number line.

So, to wrap things up: the domain of f^(-1)(x) is all real numbers except 0, and the range of f^(-1)(x) is the entire real number line, just like my range of hilarious jokes!

To find the domain and range of the inverse of the function f(x) = (1/4x) + 2, we can start by swapping the x and y variables and solving for y.

1. Swap the variables:
x = (1/4y) + 2

2. Solve for y:
Subtract 2 from both sides:
x - 2 = (1/4y)

3. Multiply both sides by 4 to isolate y:
4(x - 2) = y

Simplifying, we get:
4x - 8 = y

The resulting equation, y = 4x - 8, represents the inverse of the function f(x).

Now, let's determine the domain and range of this inverse function:

Domain: The domain of the inverse function is equal to the range of the original function f(x). Since the original function is defined for all real numbers except x = 0 (due to the denominator of 4x), the domain of the inverse function is also all real numbers except y = 0.

Domain: (-∞, 0) U (0, +∞)

Range: The original function f(x) starts at 2 and continues upwards, so the range of the inverse function will start at negative infinity and go up to 2 (but does not include 2).

Range: (-∞, 2)

To find the domain and range of the inverse of the function f(x) = (1/4x) + 2, we follow these steps:

Step 1: Start with the equation of the original function: f(x) = (1/4x) + 2.

Step 2: Swap the roles of x and y in the equation, giving us x = (1/4y) + 2.

Step 3: Now, solve the equation for y to find the equation of the inverse function.
x = (1/4y) + 2
Subtract 2 from both sides:
x - 2 = 1/4y
Multiply both sides by 4:
4(x - 2) = y
Simplify:
4x - 8 = y

Therefore, the equation of the inverse function is f^(-1)(x) = 4x - 8.

Now, let's find the domain and range of the inverse function:

Domain:
The domain of the inverse function is the range of the original function, and vice versa. Since the original function has a fraction (1/4x), we need to check if the denominator can be zero, as division by zero is undefined.

In this case, the denominator, 4x, can never be zero. Therefore, the domain of the inverse function is all real numbers (-∞, ∞).

Range:
To find the range of the inverse function, we observe that the inverse function is a linear equation in the form y = mx + b, where m is the coefficient of x and b is the y-intercept.

In our case, the coefficient of x is 4 and the y-intercept is -8. Since the coefficient of x is positive (4 > 0), this means the inverse function is an increasing function.

This indicates that the range of the inverse function is also all real numbers (-∞, ∞).

In summary:
The domain of the inverse function f^(-1)(x) = 4x - 8 is (-∞, ∞), and the range is also (-∞, ∞).

Use my answer I gave you here:

https://www.jiskha.com/questions/1760936/Inverse-of-f-x-1-4x-2