Question

Use algebraic methods to prove that the given function has an x-intercept that is equal to its y-intercept. In your final answer, include all of your calculations.

y = x/x^2 - 1

Answer 1

We have that
`y= (x)/(x^(2)-1 )`, so we can factor the denominator to get:

`y= (x)/((x-1)(x+1))`
Now, when x=0
`y= (0)/((0-1)(0+1))`, y=0; therefore the y-intercept is zero.
And, when y=0
`0= (x)/((x-1)(x+1)) = x=[0[(x-1)(x+1)]]`, x=0; therefore the x-intercept is also zero.

We can conclude that both the y-intercept and the x-intercept are equal zero.

Answer 2

Given the function, y = x/( x^2 – 1)

Add x and subtract x from (x^2 – 1) to make it a complete square

y = x/(x^2 + x – x – 1)

y = x/[x(x + 1) – 1(x + 1)]

y = x/[(x – 1)(x + 1)]

In order to find the y -intercept, we set x = 0

When x = 0

y = 0/[(0 – 1)(0 + 1)]

y = 0.

Therefore, the y-intercept is zero.

In order to find the x -intercept, we set y = 0

When y = 0

0 = x/[(x – 1)(x + 1)]

The only way the product of a division will be zero is if the numerator is zero

So, if x/[(x – 1)(x + 1)] = 0

x = 0

Therefore, the x-intercept is also zero.

Since both x- intercept and y- intercept equal zero,

Then, the given function has an x-intercept that is equal to its y-intercept