If f is function defined from R to R , is Given by F(x)= 1/2 x+5 Then find f INVERSE(x) AND THEN Find domain and range OF F INVERSE X And plot a graph
To find the inverse of f(x), we need to follow these steps:
Step 1: Replace f(x) with y.
y = (1/2)x + 5
Step 2: Swap x and y.
x = (1/2)y + 5
Step 3: Solve for y.
2x = y + 10
y = 2x - 10
So, f^(-1)(x) = 2x - 10.
To find the domain and range of f^(-1)(x), we can look at the domain and range of f(x).
For f(x) = (1/2)x + 5, the domain is all real numbers since there are no restrictions on the input x.
The range of f(x) is all real numbers as well since the graph of f(x) is a straight line that extends infinitely in both directions.
Since the inverse function, f^(-1)(x) = 2x - 10, is a linear function, it also has a domain of all real numbers.
For the range of f^(-1)(x), we can see that as x varies from negative infinity to positive infinity, the corresponding values of f^(-1)(x) will also vary from negative infinity to positive infinity. Therefore, the range of f^(-1)(x) is all real numbers as well.
To plot the graph of f^(-1)(x), we can plot a few points and then connect them with a straight line.
Let's choose x = 0:
f^(-1)(0) = 2(0) - 10 = -10
So one point is (0, -10).
Next, let's choose x = 5:
f^(-1)(5) = 2(5) - 10 = 0
So another point is (5, 0).
Plotting these two points and drawing a straight line through them will result in the graph of f^(-1)(x).
The graph of f^(-1)(x) is a straight line with a negative slope, passing through the points (0, -10) and (5, 0).
To find the inverse of the function f(x) = (1/2)x + 5, we'll follow these steps:
Step 1: Replace the function notation f(x) with y.
y = (1/2)x + 5
Step 2: Switch the x and y variables.
x = (1/2)y + 5
Step 3: Solve for y.
x - 5 = (1/2)y
2x - 10 = y
So, the inverse function of f(x) is f^(-1)(x) = 2x - 10.
Now, let's find the domain and range of the inverse function, f^(-1)(x):
Domain of f^(-1)(x) is equal to the range of f(x), which is all real numbers.
Range of f^(-1)(x) is equal to the domain of f(x), which is also all real numbers.
To plot the graph, we'll plot both f(x) and f^(-1)(x) on the same coordinate system:
Graph of f(x) = (1/2)x + 5:
- Choose some x-values.
- Calculate corresponding y-values using the given equation.
- Plot the points and draw a straight line passing through them.
Graph of f^(-1)(x) = 2x - 10:
- Choose some x-values.
- Calculate corresponding y-values using the inverse function.
- Plot the points and draw a straight line passing through them.
I hope this helps!