Find the $$ x $$-intercept(s) and $$ y $$-intercept(s), of the graph of the following. $$ y=\frac{36-x^{2}}{7+x^{2}} $$ If there is more than one answer, separate them with commas. Click on "None" If applicable.

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UpStudy AI · Accepted Answer

To find the $$ x $$-intercept(s) and $$ y $$-intercept(s) of the graph of the function

$$
y = \frac{36 - x^2}{7 + x^2},
$$

we will follow these steps:

### Finding the $$ x $$-intercept(s)

The $$ x $$-intercept(s) occur when $$ y = 0 $$. Therefore, we set the equation to zero:

$$
0 = \frac{36 - x^2}{7 + x^2}.
$$

For the fraction to be zero, the numerator must be zero (as long as the denominator is not zero). Thus, we solve:

$$
36 - x^2 = 0.
$$

Rearranging gives:

$$
x^2 = 36.
$$

Taking the square root of both sides, we find:

$$
x = 6 \quad \text{or} \quad x = -6.
$$

Thus, the $$ x $$-intercepts are:

$$
(6, 0) \quad \text{and} \quad (-6, 0).
$$

### Finding the $$ y $$-intercept(s)

The $$ y $$-intercept(s) occur when $$ x = 0 $$. We substitute $$ x = 0 $$ into the equation:

$$
y = \frac{36 - 0^2}{7 + 0^2} = \frac{36}{7}.
$$

Thus, the $$ y $$-intercept is:

$$
\left(0, \frac{36}{7}\right).
$$

### Summary of Intercepts

- The $$ x $$-intercepts are: $$ (6, 0), (-6, 0) $$.
- The $$ y $$-intercept is: $$ \left(0, \frac{36}{7}\right) $$.

### Final Answer

The $$ x $$-intercepts are: $$ 6, -6 $$.

The $$ y $$-intercept is: $$ \frac{36}{7} $$.
The $$ x $$-intercepts are: $$ 6, -6 $$. The $$ y $$-intercept is: $$ \frac{36}{7} $$.

Find the -intercept(s) and -intercept(s), of the graph of the following.

If there is more than one answer, separate them with commas.
Click on “None” If applicable.

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Find the -intercept(s) and -intercept(s), of the graph of the following. If there is more than one answer, separate them with commas. Click on “None” If applicable.