Question

Which of the following are the characteristics for the graph of the parent linear function? Select all that apply.

(1 point)
Responses

The domain and range are left-parenthesis negative infinity comma infinity right-parenthesis.
The domain and range are Image with alt text: left-parenthesis negative infinity comma infinity right-parenthesis .

It is an even function.
It is an even function.

It decreases over the interval left-parenthesis negative infinity comma infinity right-parenthesis.
It decreases over the interval Image with alt text: left-parenthesis negative infinity comma infinity right-parenthesis .

The end behavior is f left-parenthesis x right-parenthesis right-arrow infinity as x right-arrow infinity and f left-parenthesis x right-parenthesis right-arrow negative infinity as x right-arrow negative infinity.

Answer 1

To analyze the characteristics of the parent linear function, which is typically represented as \( f(x) = x \), let's evaluate each statement:

1. **The domain and range are \( (-\infty, \infty) \)**:
- This is true. The parent linear function can accept any real number as input (domain) and can produce any real number as output (range).

2. **It is an even function**:
- This is false. A function is considered even if \( f(-x) = f(x) \) for all \( x \). The parent linear function \( f(x) = x \) does not satisfy this property because \( f(-x) = -x \), which is not equal to \( f(x) \).

3. **It decreases over the interval \( (-\infty, \infty) \)**:
- This is false. The parent linear function \( f(x) = x \) is neither decreasing nor increasing across its entire domain; it is a straight line with a slope of 1, meaning it increases.

4. **The end behavior is \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \)**:
- This is true. As \( x \) increases without bound, \( f(x) = x \) goes to positive infinity, and as \( x \) decreases without bound, \( f(x) = x \) goes to negative infinity.

Based on this analysis, the correct characteristics of the graph of the parent linear function are:

- The domain and range are \( (-\infty, \infty) \).
- The end behavior is \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).