Ready In the following exercise, you will be given the opportunity to recall what happens when you add two linear functions. Given: $$ f(x)=2 x+3, g(x)=x+2 $$ $$ h(x)=x-5, m(x)=-3 x-1 $$

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Sure! Let's explore what happens when you add two linear functions using the given functions:

- $$ f(x) = 2x + 3 $$
- $$ g(x) = x + 2 $$
- $$ h(x) = x - 5 $$
- $$ m(x) = -3x - 1 $$

**Understanding Linear Functions:**
A linear function is of the form $$ y = mx + b $$, where:
- $$ m $$ is the **slope** (rate of change).
- $$ b $$ is the **y-intercept** (the value of $$ y $$ when $$ x = 0 $$).

**Adding Two Linear Functions:**
When you add two linear functions, you add their corresponding coefficients (slopes) and their constant terms (y-intercepts). The result is also a linear function.

**Step-by-Step Addition:**

1. **Adding $$ f(x) $$ and $$ g(x) $$:**
   $$
   f(x) + g(x) = (2x + 3) + (x + 2)
   $$
   - **Combine like terms:**
     - $$ 2x + x = 3x $$
     - $$ 3 + 2 = 5 $$
   $$
   f(x) + g(x) = 3x + 5
   $$
   
2. **Adding $$ h(x) $$ and $$ m(x) $$:**
   $$
   h(x) + m(x) = (x - 5) + (-3x - 1)
   $$
   - **Combine like terms:**
     - $$ x - 3x = -2x $$
     - $$ -5 - 1 = -6 $$
   $$
   h(x) + m(x) = -2x - 6
   $$
   
**Summary of Results:**
- $$ f(x) + g(x) = 3x + 5 $$
- $$ h(x) + m(x) = -2x - 6 $$

**Key Points:**
- **Slope Addition:** When adding two linear functions, their slopes add up. For example, $$ 2x $$ (from $$ f(x) $$) and $$ x $$ (from $$ g(x) $$) combine to give a slope of $$ 3x $$.
- **Y-Intercept Addition:** The constant terms add up separately. Here, $$ 3 $$ (from $$ f(x) $$) and $$ 2 $$ (from $$ g(x) $$) combine to give a constant term of $$ 5 $$.
- **Resulting Function:** The sum of two linear functions is another linear function.

**Visualization:**

1. **Graphing $$ f(x) + g(x) = 3x + 5 $$:**
   - **Slope:** 3 (steeper than $$ f(x) $$ and $$ g(x) $$)
   - **Y-Intercept:** 5

2. **Graphing $$ h(x) + m(x) = -2x - 6 $$:**
   - **Slope:** -2 (slopes downward)
   - **Y-Intercept:** -6

**Practice Problem:**
Try adding the following pairs of linear functions on your own:

- **Pair 1:**
  - $$ f(x) = 4x + 1 $$
  - $$ g(x) = -x + 3 $$
  
- **Pair 2:**
  - $$ h(x) = 5x - 2 $$
  - $$ m(x) = 2x + 4 $$
  
**Solution:**

1. **Pair 1:**
   $$
   f(x) + g(x) = (4x + 1) + (-x + 3) = (4x - x) + (1 + 3) = 3x + 4
   $$
   
2. **Pair 2:**
   $$
   h(x) + m(x) = (5x - 2) + (2x + 4) = (5x + 2x) + (-2 + 4) = 7x + 2
   $$

Feel free to ask if you have any more questions or need further explanations!
When you add two linear functions, you add their slopes and y-intercepts. For example, adding $$ f(x) = 2x + 3 $$ and $$ g(x) = x + 2 $$ gives $$ 3x + 5 $$. Similarly, adding $$ h(x) = x - 5 $$ and $$ m(x) = -3x - 1 $$ results in $$ -2x - 6 $$.

Ready In the following exercise, you will be given the opportunity to recall what happens when you add two linear functions. Given: \( f(x)=2 x+3, g(x)=x+2 \) \( h(x)=x-5, m(x)=-3 x-1 \)

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