Find the inverse of g(x)=1/3x - 7 and then find its domain and range.
Please help. Thank you.
To find the inverse of g(x) = (1/3)x - 7, we can 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 the equation for y, which means isolating y on one side of the equation:
3x = y - 21.
Step 4: Swap x and y again to get the inverse function:
y = 3x - 21.
So, the inverse of g(x) = (1/3)x - 7 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 same as the range of g(x).
The range of g(x) = (1/3)x - 7 is all real numbers since there are no restrictions on x.
Therefore, the domain of g^(-1)(x) is also all real numbers.
Range: The range of g^(-1)(x) is the same as the domain of g(x).
The domain of g(x) = (1/3)x - 7 is all real numbers.
Therefore, the range of g^(-1)(x) is also all real numbers.
So, the domain and range of the inverse function g^(-1)(x) = 3x - 21 are both all real numbers.
To find the inverse of a function, you need to swap the roles of x and y and solve for y.
Step 1: Start with the original function g(x) = 1/3x - 7.
Step 2: Swap x and y to obtain x = 1/3y - 7.
Step 3: Solve the equation for y. To do this, isolate y on one side by adding 7 to both sides of the equation:
x + 7 = 1/3y.
Step 4: Multiply both sides by 3 to get rid of the fraction:
3(x + 7) = y.
Step 5: Simplify the right side of the equation:
3x + 21 = y.
Now, we have the inverse function: g^(-1)(x) = 3x + 21.
To find the domain and range:
The domain of the inverse function (g^(-1)(x)) is the same as the range of the original function (g(x)), and the range of the inverse function is the same as the domain of the original function.
The original function g(x) = 1/3x - 7 is a linear function, which means it goes on forever in both directions. Consequently, its domain and range are both all real numbers, (-∞, ∞).
Therefore, the domain of the inverse function g^(-1)(x) is also (-∞, ∞), and its range is also all real numbers (-∞, ∞).
as written, the inverse is clearly (x+7)/3
I expect you can figure the domain and range of a sloping straight line