determine the equation in the form f(x)=1/kx-c for the function with a vertical asymptote at x=1 and a y-int at -1
vertical asymptote at x=1 ----> f(x) = 1/(x-1)
a y-int at -1 ----> f(x) = 1/(x-1) - 1
Assuming the usual sloppiness with parentheses,
f(x) = 1/(kx-c)
asymptote at x=1 means c/k = 1 so
f(x) = 1/(k(x-1))
f(0) = -1 means 1/(k(0-1)) = -1
so k = 1
f(x) = 1/(x-1)
Why did the function refuse to attend the party?
Because it knew it would be ignored at x=1 (the vertical asymptote) and was feeling a bit negative about being represented by a y-int of -1!
Now, let's find the equation of this elusive function.
Since we have a vertical asymptote at x=1, it means that there is a factor of (x-1) in the denominator of our function. So far, our equation looks like f(x) = 1/(x-1).
To find the value of k, we need to determine the factor in the numerator that will give us a y-int of -1. Since we want the y-int to be -1, we need to set the function equal to -1 when x=0:
-1 = 1/(0-1) = 1/(-1) = -1.
Perfect! We don't need to do anything more, as we have found the value of k which is 1. Our equation becomes:
f(x) = 1/x - 1.
Voila! The equation in the form f(x) = 1/kx - c, where k=1 and c=-1, is f(x) = 1/x - 1.
To determine the equation for the given function, we'll focus on the information provided:
1. Vertical asymptote at x = 1: This indicates that there is a vertical line at x = 1 where the function approaches positive or negative infinity. The equation for a vertical asymptote is x = a, where a is the value of x at the vertical asymptote.
2. y-intercept at -1: This means that the function intersects the y-axis at the point (0, -1). The y-intercept is the value of y when x = 0.
Now, let's proceed step-by-step to find the equation in the form f(x) = 1/kx - c:
Step 1: Determine the value of k:
Since the vertical asymptote is at x = 1, we know that there is a factor of (x - 1) in the denominator of the function.
Step 2: Write the equation in the form 1/kx - c:
At this point, we know that our equation has the form f(x) = 1/k(x - 1) - c.
Step 3: Find the value of c:
We know that the y-intercept is -1, which means that when x = 0, f(x) is also -1. Plugging these values into the equation gives us:
-1 = 1/k(0 - 1) - c
-1 = -1/k - c
From this equation, we can determine the value of c.
Step 4: Substitute the value of c back into the equation:
Using the value of c found in the previous step, we substitute it back into the equation:
f(x) = 1/k(x - 1) - c
Now, if you provide the value of k, we can substitute it into the equation to obtain the final form.
To determine the equation in the form f(x) = 1/kx - c for the given function, we need to use the information about the vertical asymptote and y-intercept.
First, let's consider the vertical asymptote at x = 1. A vertical asymptote occurs when the denominator of the function becomes zero. In this case, we have a vertical asymptote at x = 1, which means our denominator will have a factor of (x - 1).
Next, let's find the value of the constant k. For the function f(x) = 1/kx - c to have a vertical asymptote at x = 1, it means that k should be the reciprocal of the value in the numerator. In other words, k = 1/(value in the numerator).
Now, let's find the y-intercept, which is the value of the function when x = 0. According to the information given, the y-intercept is -1. Substituting x = 0 into the equation, we have f(0) = -1. Plugging these values into the equation, we get:
f(0) = 1/k(0) - c
-1 = 0 - c
c = 1
Finally, we have determined the values of k and c. The equation in the form f(x) = 1/kx - c, with a vertical asymptote at x = 1 and a y-intercept at -1, is:
f(x) = 1/x - 1