For each function,
i) determine the corresponding reciprocal
function, y =1/f(x)
ii) state the non-permissible values of
x and the equation(s) of the vertical
asymptote(s) of the reciprocal function
iii) determine the x-intercepts and the
y-intercept of the reciprocal function
iv) sketch the graphs of y = f(x) and
y = 1/f(x) on the same set of axes
a) f(x) = x^2 - 25
b) f(x) = x^2 - 6x + 5
I will do the first, you do the second
f(x) = x^2 - 25
let g(x) = 1/(x^2 - 25)
graphs: I assume you know what the parabola looks like, here is the reciprocal
http://www.wolframalpha.com/input/?i=plot++y+%3D+1%2F%28x%5E2+-+25%29
notice that at the x-intercepts of the first one, namely x=±5, the reciprocal has asymptotes,
so vertical asymptotes at x = ± 5
there are no x-intercepts, (since 1/(x^-25) can never be zero.
for the y-intercept , let x = 0
so 1/(0-25) = -.04 (see graph)
Thank you so much, the second one is easy now :)
To find the corresponding reciprocal function, let's follow these steps for each given function:
a) f(x) = x^2 - 25
i) The reciprocal function is y = 1/f(x). So, the reciprocal of f(x) is y = 1/(x^2 - 25).
ii) In this case, we need to identify the non-permissible values of x and the equations of the vertical asymptotes for the reciprocal function.
To find the non-permissible values, we set the denominator of the reciprocal function equal to zero and solve for x:
x^2 - 25 = 0
(x - 5)(x + 5) = 0
x = 5 and x = -5
So, the non-permissible values for x are 5 and -5.
To find the equations of the vertical asymptotes, we can use these non-permissible values. Therefore, the equations of the vertical asymptotes are x = 5 and x = -5.
iii) To determine the x-intercepts of the reciprocal function, we set y = 0 and solve for x:
1/(x^2 - 25) = 0
Since a fraction is equal to zero only when the numerator is zero, we can ignore the denominator:
x^2 - 25 = 0
(x - 5)(x + 5) = 0
x = 5 and x = -5
So, the x-intercepts of the reciprocal function are x = 5 and x = -5.
To find the y-intercept, we set x = 0 in the reciprocal function:
y = 1/(0^2 - 25)
y = -1/25
So, the y-intercept of the reciprocal function is y = -1/25.
iv) Now we can sketch the graphs of y = f(x) and y = 1/f(x) on the same set of axes. Here is a graph that represents both functions:
(please note that the graph visualization cannot be created by text-based format)
b) f(x) = x^2 - 6x + 5
Follow the same steps as above to find the corresponding reciprocal function, non-permissible values of x, equations of vertical asymptotes, x-intercepts, and y-intercept for the function f(x) = x^2 - 6x + 5.