OPERATIONS ON FUNCTIONS

• g(x) = x ^ 2 + 2x - 8

• f(x) = x + 3

•h(x) = (x + 7)/(2 - x)

• p(x) = 2x - 7

•t(x) = (x - x)/(x + 3)

• v(x) = x ^ 2 + 5x + 4

f(x) t t(x)
v(x) - h(x)
p(x) t h(x)
g(x) - t(x)
f(x) x t(x)
h(x) ÷ v(x)
p(x) x h(x)

Let's evaluate the expressions based on the operations and the given functions:

### Given Functions:
1. \( g(x) = x^2 + 2x - 8 \)
2. \( f(x) = x + 3 \)
3. \( h(x) = \frac{x + 7}{2 - x} \)
4. \( p(x) = 2x - 7 \)
5. \( t(x) = \frac{x - x}{x + 3} \)
6. \( v(x) = x^2 + 5x + 4 \)

### 1. \( f(x) \cdot t(x) \)
\[
f(x) \cdot t(x) = (x + 3) \cdot \frac{x - x}{x + 3} = (x + 3) \cdot 0 = 0
\]
Since \( t(x) = \frac{0}{x + 3} = 0 \), the result of this operation is \( 0 \).

### 2. \( v(x) - h(x) \)
\[
v(x) - h(x) = \left(x^2 + 5x + 4\right) - \frac{x + 7}{2 - x}
\]
This expression can be rewritten as:
\[
v(x) - h(x) = x^2 + 5x + 4 - \frac{x + 7}{2 - x}
\]
You would need to find a common denominator to combine these, but this is the general form of the difference.

### 3. \( p(x) \cdot h(x) \)
\[
p(x) \cdot h(x) = (2x - 7) \cdot \frac{x + 7}{2 - x}
\]
This becomes:
\[
p(x) \cdot h(x) = \frac{(2x - 7)(x + 7)}{2 - x}
\]
You can expand the numerator using the distributive property:
\[
(2x - 7)(x + 7) = 2x^2 + 14x - 7x - 49 = 2x^2 + 7x - 49
\]
Thus,
\[
p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}
\]

### 4. \( g(x) - t(x) \)
Since \( t(x) = 0 \),
\[
g(x) - t(x) = g(x) - 0 = g(x) = x^2 + 2x - 8
\]

### 5. \( f(x) \cdot t(x) \)
As we saw in the first calculation, \( f(x) \cdot t(x) = 0 \).

### 6. \( h(x) \div v(x) \)
\[
\frac{h(x)}{v(x)} = \frac{\frac{x + 7}{2 - x}}{x^2 + 5x + 4}
\]
This can be rewritten as:
\[
\frac{x + 7}{(2 - x)(x^2 + 5x + 4)}
\]
You could simplify further, depending on the specific values of \( x \).

### 7. \( p(x) \cdot h(x) \)
This was already evaluated in step 3:
\[
p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}
\]

### Summary of Results:
1. \( f(x) \cdot t(x) = 0 \)
2. \( v(x) - h(x) = x^2 + 5x + 4 - \frac{x + 7}{2 - x} \)
3. \( p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x} \)
4. \( g(x) - t(x) = x^2 + 2x - 8 \)
5. \( f(x) \cdot t(x) = 0 \)
6. \( \frac{h(x)}{v(x)} = \frac{x + 7}{(2 - x)(x^2 + 5x + 4)} \)
7. \( p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x} \)

Do you have any questions or would you like further details on any step?

### Related Questions:
1. What happens when you simplify \( v(x) - h(x) \)?
2. How does the form of \( p(x) \cdot h(x) \) change when you simplify further?
3. What is the behavior of these functions for different values of \( x \)?
4. How do you find the domain restrictions for \( h(x) \) and \( v(x) \)?
5. What would \( f(x) \div t(x) \) look like if \( t(x) \) were not zero?

### Tip:
When dealing with fractions or rational expressions, always check for possible restrictions on the domain, especially when a denominator can be zero.

OPERATIONS ON FUNCTIONS

• g(x) = x^2 + 2x - 8
• f(x) = x + 3
• h(x) = (x + 7)/(2 - x)
• p(x) = 2x - 7
• t(x) = (x - x)/(x + 3)
• v(x) = x^2 + 5x + 4

f(x) t t(x)
v(x) - h(x)
p(x) t h(x)
g(x) - t(x)
f(x) x t(x)
h(x) ÷ v(x)
p(x) x h(x)

Explore operations on functions, including multiplication, subtraction, and division, with detailed examples. Functions involved are quadratic, linear, and rational expressions. Perfect for high school students in Grades 9-12.

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