MATH SELFIE

A. Perform the following fundamental operations on each given functions.Assignment: Answer 1, 3, 5, 7, 9. Refer to c.

c. (f•g)(x)

1. f(x) = x + 1 g(x) = 2x - 3

3. f(x) = x ^ 2 + 1 g(x) = x - 3

5. f(x) = x - 2 g(x) = 1/x

7. f(x) = 3x ^ 2 + 1; g(x) = x - 3

9. f(x) = 2/x g(x) = 1/x

To perform the operation \((f \cdot g)(x)\), which represents the composition of two functions, we evaluate \(g(x)\) first and then substitute that result into \(f(x)\). This is different from the product of the functions \(f(x) \times g(x)\). Here’s how you can calculate \((f \cdot g)(x)\) for each of the given pairs of functions:

### 1. \( f(x) = x + 1 \), \( g(x) = 2x - 3 \)
\[
(f \cdot g)(x) = f(g(x)) = f(2x - 3)
\]
Substitute \(2x - 3\) into \(f(x)\):
\[
f(2x - 3) = (2x - 3) + 1 = 2x - 2
\]
So, \((f \cdot g)(x) = 2x - 2\).

### 3. \( f(x) = x^2 + 1 \), \( g(x) = x - 3 \)
\[
(f \cdot g)(x) = f(g(x)) = f(x - 3)
\]
Substitute \(x - 3\) into \(f(x)\):
\[
f(x - 3) = (x - 3)^2 + 1
\]
Expand \((x - 3)^2\):
\[
(x - 3)^2 = x^2 - 6x + 9
\]
So,
\[
f(x - 3) = x^2 - 6x + 9 + 1 = x^2 - 6x + 10
\]
Thus, \((f \cdot g)(x) = x^2 - 6x + 10\).

### 5. \( f(x) = x - 2 \), \( g(x) = \frac{1}{x} \)
\[
(f \cdot g)(x) = f\left(\frac{1}{x}\right)
\]
Substitute \(\frac{1}{x}\) into \(f(x)\):
\[
f\left(\frac{1}{x}\right) = \frac{1}{x} - 2
\]
So, \((f \cdot g)(x) = \frac{1}{x} - 2\).

### 7. \( f(x) = 3x^2 + 1 \), \( g(x) = x - 3 \)
\[
(f \cdot g)(x) = f(g(x)) = f(x - 3)
\]
Substitute \(x - 3\) into \(f(x)\):
\[
f(x - 3) = 3(x - 3)^2 + 1
\]
Expand \((x - 3)^2\):
\[
(x - 3)^2 = x^2 - 6x + 9
\]
So,
\[
f(x - 3) = 3(x^2 - 6x + 9) + 1 = 3x^2 - 18x + 27 + 1 = 3x^2 - 18x + 28
\]
Thus, \((f \cdot g)(x) = 3x^2 - 18x + 28\).

### 9. \( f(x) = \frac{2}{x} \), \( g(x) = \frac{1}{x} \)
\[
(f \cdot g)(x) = f\left(\frac{1}{x}\right)
\]
Substitute \(\frac{1}{x}\) into \(f(x)\):
\[
f\left(\frac{1}{x}\right) = \frac{2}{\frac{1}{x}} = 2x
\]
So, \((f \cdot g)(x) = 2x\).

### Summary of Answers:
1. \((f \cdot g)(x) = 2x - 2\)
3. \((f \cdot g)(x) = x^2 - 6x + 10\)
5. \((f \cdot g)(x) = \frac{1}{x} - 2\)
7. \((f \cdot g)(x) = 3x^2 - 18x + 28\)
9. \((f \cdot g)(x) = 2x\)

Would you like further details or have any questions?  

### Further Questions:
1. How do you differentiate between function composition and multiplication?
2. What is the difference between \((f \cdot g)(x)\) and \((g \cdot f)(x)\)?
3. How do you expand a binomial like \((x - 3)^2\)?
4. What happens if \(g(x)\) is a constant function?
5. Can function composition be applied to trigonometric functions?
6. What is the significance of the order in function composition?
7. How do you interpret the result of \((f \cdot g)(x)\) graphically?
8. How can you verify if a function is invertible before composition?

**Tip:** When composing functions, always ensure that the output of the inner function (g(x)) falls within the domain of the outer function (f(x)).

Explore how to perform function composition (f • g)(x) for various pairs of functions. This problem involves evaluating compositions such as (2x - 3) + 1, (x - 3)^2 + 1, 1/x - 2, 3(x - 3)^2 + 1, and 2x. Suitable for high school students learning advanced algebraic operations with functions.

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