What is the domain and range of  y=1+x/5-x

The function given is:

\[
y = \frac{1 + x}{5 - x}
\]

### Domain:
The domain of a function consists of all values of \( x \) for which the function is defined. In this case, the function is undefined when the denominator equals zero. To find when the denominator is zero:

\[
5 - x = 0
\]

\[
x = 5
\]

Thus, the function is undefined at \( x = 5 \). Therefore, the domain is all real numbers except \( x = 5 \). In interval notation, the domain is:

\[
\text{Domain} = (-\infty, 5) \cup (5, \infty)
\]

### Range:
The range of the function refers to all possible values of \( y \). To find the range, let's manipulate the equation and solve for \( x \) in terms of \( y \):

\[
y = \frac{1 + x}{5 - x}
\]

Multiply both sides by \( 5 - x \):

\[
y(5 - x) = 1 + x
\]

Distribute \( y \):

\[
5y - xy = 1 + x
\]

Group terms involving \( x \) on one side:

\[
5y - 1 = x + xy
\]

Factor out \( x \) on the right-hand side:

\[
5y - 1 = x(1 + y)
\]

Now, solve for \( x \):

\[
x = \frac{5y - 1}{1 + y}
\]

For \( x \) to be defined, the denominator \( 1 + y \) cannot be zero, which gives \( y = -1 \). Therefore, the function is undefined when \( y = -1 \).

Thus, the range is all real numbers except \( y = -1 \). In interval notation, the range is:

\[
\text{Range} = (-\infty, -1) \cup (-1, \infty)
\]

### Summary:
- **Domain**: \( (-\infty, 5) \cup (5, \infty) \)
- **Range**: \( (-\infty, -1) \cup (-1, \infty) \)

Would you like further details or have any questions?

Here are some related questions you might want to explore:
1. How can the vertical asymptote of this function be determined?
2. What happens to \( y \) as \( x \to \infty \)?
3. What is the horizontal asymptote of this function?
4. How would you find the intercepts of this function?
5. What is the behavior of this function around \( x = 5 \)?

**Tip**: The vertical asymptote occurs where the function's denominator is zero.

What is the domain and range of y = 1 + x / 5 - x?

Discover how to find the domain and range of the rational function y = (1 + x) / (5 - x). This solution explains the steps to identify when the function is undefined and how to solve for x in terms of y to determine the range. Suitable for students in Grades 9-12.

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