Explain the interval(s) where the function f(x)= _1/2x + 10
a) positive
b) increasing is

Let's analyze the function \( f(x) = \frac{1}{2}x + 10 \).

### a) Where is the function positive?

The function is positive when \( f(x) > 0 \).

\[
f(x) = \frac{1}{2}x + 10 > 0
\]

To find where this inequality holds, solve for \( x \):

\[
\frac{1}{2}x + 10 > 0
\]

Subtract 10 from both sides:

\[
\frac{1}{2}x > -10
\]

Multiply both sides by 2:

\[
x > -20
\]

So, the function is positive for \( x > -20 \).

**Answer**: The function is positive on the interval \( (-20, \infty) \).

### b) Where is the function increasing?

For linear functions like \( f(x) = \frac{1}{2}x + 10 \), the slope of the line determines whether the function is increasing or decreasing. Since the coefficient of \( x \) is \( \frac{1}{2} \), which is positive, the function is always increasing.

**Answer**: The function is increasing on the interval \( (-\infty, \infty) \).

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Would you like more details or have any questions? Here are some related questions:

1. How do we determine the intervals where a function is negative?
2. What is the interval where the function \( f(x) = -\frac{1}{2}x + 10 \) is positive?
3. How do we find where a quadratic function is increasing or decreasing?
4. Can a function be both increasing and decreasing on the same interval?
5. How does the slope of a linear function affect its graph?

**Tip:** For linear functions, the sign of the slope determines if the function is increasing (positive slope) or decreasing (negative slope).

Explain the interval(s) where the function f(x)= 1/2x + 10 is a) positive b) increasing.

Discover how to determine the intervals where the linear function f(x) = 1/2x + 10 is positive and increasing. This problem involves key algebraic concepts such as linear functions and inequalities. Ideal for students in Grades 8-10.

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