find the inverse of g(x)=1/3x-7 and find its domain and range
To find the inverse of a function, we need to interchange the roles of x and y and solve for y.
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
y = 3x + 21
Therefore, the inverse of g(x) = (1/3)x - 7 is g^(-1)(x) = 3x + 21.
The domain of g(x) is all real numbers since there are no restrictions on the value of x.
The range is also all real numbers since the slope of the function is 1/3, which means the graph of g(x) is always increasing. Therefore, for any real number x, g(x) will produce a unique real number y.
To find the inverse of the function g(x) = 1/3x - 7, we need to interchange x and y and then solve for y.
Step 1: Interchange x and y:
x = 1/3y - 7
Step 2: Solve for y:
Multiply both sides of the equation by 3 to eliminate the fraction:
3x = y - 21
Add 21 to both sides of the equation:
3x + 21 = y
So, the inverse of g(x) is g^(-1)(x) = 3x + 21.
Now, let's find the domain and range of the inverse function.
Domain:
The domain of g^(-1)(x) is the set of all real numbers because there are no restrictions on the value of x in the equation.
Range:
The range of g^(-1)(x) is also the set of all real numbers because the equation y = 3x + 21 can take on any value of y for any given value of x.
Therefore, the domain of the inverse function g^(-1)(x) is (-∞, ∞), and the range is also (-∞, ∞).
To find the inverse of a function, you need to follow these steps:
Step 1: Replace the given function notation, g(x), with y.
y = 1/3x - 7
Step 2: Swap the roles of x and y in the equation.
x = 1/3y - 7
Step 3: Solve for y.
x + 7 = 1/3y
3(x + 7) = y
3x + 21 = y
Now, replace y with its function notation inverse, g^(-1)(x):
g^(-1)(x) = 3x + 21
The inverse of g(x) is g^(-1)(x) = 3x + 21.
For the domain and range of the inverse function, we need to consider the domain and range of the original function.
The original function, g(x) = 1/3x - 7, has a domain of all real numbers because there are no restrictions on the x-values.
To find the range of g(x), we can see that the coefficient of x in the function is positive (1/3), which means the graph of the function opens upward. As there are no restrictions on the x-values, the range will also be all real numbers.
Since the inverse function swaps the domain and range of the original function, the domain of the inverse function, g^(-1)(x), will be all real numbers. Similarly, the range of the inverse function will also be all real numbers.