Find the inverse function f^-1 of the function f. What is the domain of f^-1? What is its range?
f(x) = 5 tan (10x-2)
switch variables and solve for y:
x = 5tan(10y-2)
x/5 = tan(10y-2)
10y-2 = arctan(x/5)
y = (arctan(x/5)+2)/10
In order for tan(z) to have an inverse, -π/2 < z < π/2
So, we need to restrict f(x) so its domain is
-π/2 < 10x-2 < π/2
2-π/2 < 10x < 2+π/2
(2-π/2)/10 < x < (2+π/2)/10
Now you can see what the domain and range of the f^-1 are.
To find the inverse function f^-1, we need to switch the roles of x and f(x) in the function f, and solve for f^-1(x).
Let's start by switching x and f(x):
x = 5 tan(10f^-1(x) - 2)
Next, let's solve for f^-1(x):
Divide both sides of the equation by 5:
x/5 = tan(10f^-1(x) - 2)
Take the arctangent (inverse tangent) of both sides:
arctan(x/5) = 10f^-1(x) - 2
Add 2 to both sides:
arctan(x/5) + 2 = 10f^-1(x)
Divide both sides by 10:
(arctan(x/5) + 2)/10 = f^-1(x)
Therefore, the inverse function f^-1(x) is:
f^-1(x) = (arctan(x/5) + 2)/10
Now, let's determine the domain and range of f^-1:
The domain of f^-1 consists of all the possible input values (x-values) for which the inverse function is defined. In this case, we can take any real number as input for f^-1(x).
Therefore, the domain of f^-1 is the set of all real numbers.
The range of f^-1 consists of all the possible output values (y-values) for the inverse function. The arctangent function has a range of (-π/2, π/2), and adding 2 and dividing by 10 will still keep the range between -π/2 and π/2.
Therefore, the range of f^-1 is (-π/2, π/2).
To find the inverse function, f^(-1), you need to perform the following steps:
Step 1: Replace f(x) with y: y = 5 tan(10x - 2).
Step 2: Swap x and y: x = 5 tan(10y - 2).
Step 3: Solve the equation for y. In this case, we have a trigonometric function, so we'll use inverse trigonometric methods:
x = 5 tan(10y - 2)
tan^(-1)(x/5) = 10y - 2
10y = tan^(-1)(x/5) + 2
y = (tan^(-1)(x/5) + 2) / 10
Therefore, the inverse function is: f^(-1)(x) = (tan^(-1)(x/5) + 2) / 10.
Now, let's determine the domain and range of f^(-1):
The domain of f^(-1) is the set of all possible input values for x that will yield real output values. In this case, since we are using the inverse of the tangent function, the domain of f^(-1) is all real numbers.
The range of f^(-1) is the set of all possible output values for y. As we are using the inverse of the tangent function, the range is also all real numbers, from negative infinity to positive infinity.