Find the inverse of g(x)=(1/3x)-7 and then find its domain and range.
I already found the inverse of the function.
y + 7 = 1 / 3x
reverse x and y
x + 7 = 1/3y
3y = 1/(x+7)
y = 1/(3x+21) that is what I get for inverse.
domain is all real numbers EXCEPT x = -7
range is all real numbers (look at it going to zero at x big - and at x big + and to - and + infinity as x approaches -7
Well, well, congratulations on finding the inverse! You must have been doing some serious brain gymnastics. Now, let's dive in and find the domain and range.
First, let's remind ourselves that the domain refers to the set of all possible values that x can take, and the range refers to the set of all possible values that y can take.
For the inverse function, let's call it h(x), we have:
h(x) = (1/3x) - 7
Now, let's tackle the domain. Since the original function had a fraction, we need to make sure that the denominator, 3x, is not zero. Therefore, x cannot be 0, as it would make the denominator zero, which is a big no-no in the mathematical world. So, the domain is all real numbers except 0.
As for the range, well, the denominator is also a factor here. Since we want to avoid division by zero, the range cannot include the number 0. The rest of the numbers are fair game, from negative infinity all the way to positive infinity!
So, to sum it up:
Domain: All real numbers excluding 0.
Range: All real numbers excluding 0.
Hope that clears things up! If you need any more mathematical hilarity or help, just give me a shout!
To find the inverse of the function g(x) = (1/3)x - 7, we need to swap the variable x and y and solve for y. Let's proceed step-by-step:
Step 1: Replace g(x) with y:
y = (1/3)x - 7
Step 2: Swap x and y:
x = (1/3)y - 7
Step 3: Solve for y:
3x = y - 21
3x + 21 = y
So, the inverse function of g(x) is g^(-1)(x) = 3x + 21.
Now let's find the domain and range of the inverse function, g^(-1)(x) = 3x + 21.
The domain of the inverse function is the set of all possible input values (x-values) for the function. Since the inverse function is a linear function, there are no restrictions on the x-values. Therefore, the domain is (-∞, ∞), which means all real numbers.
The range of the function is the set of all possible output values (y-values) for the function. Since the inverse function is a linear function, the range is also all real numbers, which can be represented as (-∞, ∞).
To find the inverse of the function g(x)=(1/3)x-7, follow these steps:
Step 1: Replace g(x) with y: y = (1/3)x - 7.
Step 2: Swap x and y: x = (1/3)y - 7.
Step 3: Solve for y: Multiply both sides by 3 to eliminate the fraction: 3x = y - 21.
Step 4: Add 21 to both sides: y = 3x + 21.
Therefore, the inverse of g(x) is f(x) = 3x + 21.
Now, let's find the domain and range of the inverse function f(x) = 3x + 21:
Domain: The domain refers to all possible input values for which the function is defined. In this case, there are no restrictions on the values of x. So, the domain of f(x) is (-∞, ∞), which means all real numbers.
Range: The range refers to all possible output values of the function. Since f(x) = 3x + 21 is a linear function, the range is also all real numbers. Therefore, the range of f(x) is (-∞, ∞).
In summary:
Domain of f(x): (-∞, ∞) - All real numbers
Range of f(x): (-∞, ∞) - All real numbers