Let f(x)=1/ax+b where and b are non zero constants. Find all solutions to f^-1(x)=0. Express your answer in terms of a and/or b. Thank you.
To find the solutions to f^(-1)(x) = 0, we need to determine the value(s) of x that make f^(-1)(x) equal to zero.
First, recall that the inverse function f^(-1)(x) reverses the roles of x and f(x). In other words, if (x, y) is a point on the graph of f(x), then (y, x) is a point on the graph of f^(-1)(x).
Therefore, f^(-1)(x) = 0 means that the y-coordinate (or output) of the inverse function is zero. To find these points, we need to solve for x when f(x) = 0.
Given that f(x) = 1/(ax + b), we can set f(x) equal to zero:
0 = 1/(ax + b)
To solve for x, we can multiply both sides of the equation by (ax + b) to get rid of the denominator:
0 * (ax + b) = 1
0 = 1
This implies that there are no solutions for x that makes f^(-1)(x) equal to zero.
Therefore, there are no solutions to f^(-1)(x) = 0 in terms of a and/or b.
To find the solutions to f^(-1)(x) = 0, we need to first find the inverse function of f(x).
Given that f(x) = (1/ax) + b, we can start by replacing f(x) with y:
y = (1/ax) + b
Next, we can swap the x and y variables:
x = (1/ay) + b
Now, let's solve this equation for y:
x - b = 1/ay
To isolate y, we can take the reciprocal of both sides:
1/(x - b) = ay
Finally, divide both sides of the equation by a:
y = 1/(a(x - b))
Therefore, the inverse function of f(x) is f^(-1)(x) = 1/(a(x - b)).
Now, we can find the solutions to f^(-1)(x) = 0 by setting y = 0 in the inverse function:
0 = 1/(a(x - b))
To solve for x, we can take the reciprocal of both sides:
1/0 = a(x - b)
Since the reciprocal of zero is undefined, we conclude that there are no solutions to f^(-1)(x) = 0.
Hence, the expression in terms of a and b representing all solutions to f^(-1)(x) = 0 is empty, indicating that there are no solutions.