Ai add or subtract of the following functions

f(x) = x+5 
g (x) = 4x+3
h(x) = x^2+7x+10
m(x) = x+2/x+5

1. (f+g) (x)
2. (h+g) (-3)
3. (h+f) (0)
4. (g-f( (5)
5. (h-g) (2)

Let's tackle each of these operations step by step:

### 1. \( (f+g)(x) \)
This represents the sum of the functions \( f(x) \) and \( g(x) \).

\[
f(x) = x + 5
\]
\[
g(x) = 4x + 3
\]

\[
(f + g)(x) = f(x) + g(x) = (x + 5) + (4x + 3) = 5x + 8
\]

So, \( (f+g)(x) = 5x + 8 \).

### 2. \( (h+g)(-3) \)
Here, you first find \( h(x) + g(x) \) and then evaluate it at \( x = -3 \).

\[
h(x) = x^2 + 7x + 10
\]
\[
g(x) = 4x + 3
\]

\[
(h + g)(x) = h(x) + g(x) = (x^2 + 7x + 10) + (4x + 3) = x^2 + 11x + 13
\]

Now, substitute \( x = -3 \):

\[
(h + g)(-3) = (-3)^2 + 11(-3) + 13 = 9 - 33 + 13 = -11
\]

So, \( (h+g)(-3) = -11 \).

### 3. \( (h+f)(0) \)
First, find \( h(x) + f(x) \) and then evaluate it at \( x = 0 \).

\[
h(x) = x^2 + 7x + 10
\]
\[
f(x) = x + 5
\]

\[
(h + f)(x) = h(x) + f(x) = (x^2 + 7x + 10) + (x + 5) = x^2 + 8x + 15
\]

Now, substitute \( x = 0 \):

\[
(h + f)(0) = 0^2 + 8(0) + 15 = 15
\]

So, \( (h+f)(0) = 15 \).

### 4. \( (g-f)(5) \)
This represents the difference between \( g(x) \) and \( f(x) \), evaluated at \( x = 5 \).

\[
g(x) = 4x + 3
\]
\[
f(x) = x + 5
\]

\[
(g - f)(x) = g(x) - f(x) = (4x + 3) - (x + 5) = 3x - 2
\]

Now, substitute \( x = 5 \):

\[
(g - f)(5) = 3(5) - 2 = 15 - 2 = 13
\]

So, \( (g-f)(5) = 13 \).

### 5. \( (h-g)(2) \)
This represents the difference between \( h(x) \) and \( g(x) \), evaluated at \( x = 2 \).

\[
h(x) = x^2 + 7x + 10
\]
\[
g(x) = 4x + 3
\]

\[
(h - g)(x) = h(x) - g(x) = (x^2 + 7x + 10) - (4x + 3) = x^2 + 3x + 7
\]

Now, substitute \( x = 2 \):

\[
(h - g)(2) = 2^2 + 3(2) + 7 = 4 + 6 + 7 = 17
\]

So, \( (h-g)(2) = 17 \).

---

### Summary of Results:
1. \( (f+g)(x) = 5x + 8 \)
2. \( (h+g)(-3) = -11 \)
3. \( (h+f)(0) = 15 \)
4. \( (g-f)(5) = 13 \)
5. \( (h-g)(2) = 17 \)

Do you want details on any of these steps or have any questions?

---

### Relative Questions:
1. What is \( (f \times g)(x) \) for the given functions?
2. How do you find the derivative of \( (h-f)(x) \)?
3. Can you evaluate \( (h/f)(1) \)?
4. What would \( (m + g)(x) \) look like if \( m(x) = \frac{x+2}{x+5} \)?
5. How does the addition of two functions change the graph of the resulting function?

### Tip:
When adding or subtracting functions, always combine like terms carefully to avoid errors.

Ai add or subtract of the following functions: f(x) = x+5, g(x) = 4x+3, h(x) = x^2+7x+10, m(x) = x+2/x+5. 1. (f+g)(x), 2. (h+g)(-3), 3. (h+f)(0), 4. (g-f)(5), 5. (h-g)(2)

Explore the process of adding and subtracting polynomial and rational functions with examples like f(x) = x+5, g(x) = 4x+3, h(x) = x^2+7x+10, and m(x) = x+2/x+5. Learn how to solve specific function operations and evaluate them at given points, suitable for students in Grades 9-11.

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