Given: f(x)=¾×-3 ; g(x)= ⅖×+⅔       Find: 1. (f+g)³ 2.(f-g)³ 3.(f•g)³ 4.(f/g)³ 5.(f o g)³ 6.(g o f)³

Let's break down the problem step by step to find the required expressions.

### Given Functions:
1. \( f(x) = \frac{3}{4}x - 3 \)
2. \( g(x) = \frac{2}{5}x + \frac{2}{3} \)

### Step 1: Find \( (f+g)(x) \)
\[
(f+g)(x) = f(x) + g(x) = \left(\frac{3}{4}x - 3\right) + \left(\frac{2}{5}x + \frac{2}{3}\right)
\]
Combine the like terms:
\[
(f+g)(x) = \left(\frac{3}{4}x + \frac{2}{5}x\right) + \left(-3 + \frac{2}{3}\right)
\]
To combine the coefficients of \(x\), find the common denominator:
\[
\frac{3}{4}x = \frac{15}{20}x \quad \text{and} \quad \frac{2}{5}x = \frac{8}{20}x
\]
So,
\[
(f+g)(x) = \frac{23}{20}x + \left(-3 + \frac{2}{3}\right)
\]
For the constants, find the common denominator:
\[
-3 = -\frac{9}{3} \quad \text{so} \quad -3 + \frac{2}{3} = -\frac{9}{3} + \frac{2}{3} = -\frac{7}{3}
\]
Thus,
\[
(f+g)(x) = \frac{23}{20}x - \frac{7}{3}
\]

Now we find \( (f+g)^3 \):
\[
(f+g)^3 = \left(\frac{23}{20}x - \frac{7}{3}\right)^3
\]

### Step 2: Find \( (f-g)(x) \)
\[
(f-g)(x) = f(x) - g(x) = \left(\frac{3}{4}x - 3\right) - \left(\frac{2}{5}x + \frac{2}{3}\right)
\]
Combine the like terms:
\[
(f-g)(x) = \left(\frac{3}{4}x - \frac{2}{5}x\right) + \left(-3 - \frac{2}{3}\right)
\]
Using the same steps as above to combine:
\[
(f-g)(x) = \frac{15}{20}x - \frac{8}{20}x = \frac{7}{20}x
\]
For the constants:
\[
-3 - \frac{2}{3} = -\frac{9}{3} - \frac{2}{3} = -\frac{11}{3}
\]
So,
\[
(f-g)(x) = \frac{7}{20}x - \frac{11}{3}
\]

Now we find \( (f-g)^3 \):
\[
(f-g)^3 = \left(\frac{7}{20}x - \frac{11}{3}\right)^3
\]

### Step 3: Find \( (f \cdot g)(x) \)
\[
(f \cdot g)(x) = f(x) \cdot g(x) = \left(\frac{3}{4}x - 3\right) \cdot \left(\frac{2}{5}x + \frac{2}{3}\right)
\]
Expand the product:
\[
(f \cdot g)(x) = \frac{3}{4}x \cdot \frac{2}{5}x + \frac{3}{4}x \cdot \frac{2}{3} - 3 \cdot \frac{2}{5}x - 3 \cdot \frac{2}{3}
\]
Calculate each term:
\[
\frac{3}{4}x \cdot \frac{2}{5}x = \frac{6}{20}x^2 = \frac{3}{10}x^2
\]
\[
\frac{3}{4}x \cdot \frac{2}{3} = \frac{6}{12}x = \frac{1}{2}x
\]
\[
-3 \cdot \frac{2}{5}x = -\frac{6}{5}x
\]
\[
-3 \cdot \frac{2}{3} = -2
\]
Combine the terms:
\[
(f \cdot g)(x) = \frac{3}{10}x^2 + \left(\frac{1}{2}x - \frac{6}{5}x\right) - 2
\]
Combine the linear terms:
\[
\frac{1}{2}x - \frac{6}{5}x = \frac{5}{10}x - \frac{12}{10}x = -\frac{7}{10}x
\]
So,
\[
(f \cdot g)(x) = \frac{3}{10}x^2 - \frac{7}{10}x - 2
\]

Now we find \( (f \cdot g)^3 \):
\[
(f \cdot g)^3 = \left(\frac{3}{10}x^2 - \frac{7}{10}x - 2\right)^3
\]

### Step 4: Find \( (f/g)(x) \)
\[
(f/g)(x) = \frac{f(x)}{g(x)} = \frac{\frac{3}{4}x - 3}{\frac{2}{5}x + \frac{2}{3}}
\]
Simplify this expression:
\[
(f/g)(x) = \frac{\frac{3}{4}x - 3}{\frac{2}{5}x + \frac{2}{3}}
\]

Now we find \( (f/g)^3 \):
\[
(f/g)^3 = \left(\frac{\frac{3}{4}x - 3}{\frac{2}{5}x + \frac{2}{3}}\right)^3
\]

### Step 5: Find \( (f \circ g)(x) \)
\[
(f \circ g)(x) = f(g(x)) = f\left(\frac{2}{5}x + \frac{2}{3}\right)
\]
Substitute \( g(x) \) into \( f(x) \):
\[
f\left(\frac{2}{5}x + \frac{2}{3}\right) = \frac{3}{4}\left(\frac{2}{5}x + \frac{2}{3}\right) - 3
\]
Distribute:
\[
f\left(\frac{2}{5}x + \frac{2}{3}\right) = \frac{6}{20}x + \frac{6}{12} - 3 = \frac{3}{10}x + \frac{1}{2} - 3
\]
Simplify:
\[
f\left(\frac{2}{5}x + \frac{2}{3}\right) = \frac{3}{10}x - \frac{5}{2}
\]

Now we find \( (f \circ g)^3 \):
\[
(f \circ g)^3 = \left(\frac{3}{10}x - \frac{5}{2}\right)^3
\]

### Step 6: Find \( (g \circ f)(x) \)
\[
(g \circ f)(x) = g(f(x)) = g\left(\frac{3}{4}x - 3\right)
\]
Substitute \( f(x) \) into \( g(x) \):
\[
g\left(\frac{3}{4}x - 3\right) = \frac{2}{5}\left(\frac{3}{4}x - 3\right) + \frac{2}{3}
\]
Distribute:
\[
g\left(\frac{3}{4}x - 3\right) = \frac{6}{20}x - \frac{6}{5} + \frac{2}{3}
\]
Simplify:
\[
g\left(\frac{3}{4}x - 3\right) = \frac{3}{10}x - \frac{18}{15} + \frac{2}{3} = \frac{3}{10}x - \frac{8}{15}
\]

Now we find \( (g \circ f)^3 \):
\[
(g \circ f)^3 = \left(\frac{3}{10}x - \frac{8}{15}\right)^3
\]

### Final Answers:
1. \( (f+g)^3 = \left(\frac{23}{20}x - \frac{7}{3}\right)^3 \)
2. \( (f-g)^3 = \left(\frac{7}{20}x - \frac{11}{3}\right)^3 \)
3. \( (f \cdot g)^3 = \left(\frac{3}{10}x^2 - \frac{7}{10}x

Learn how to solve operations on functions (f+g)^3, (f-g)^3, (f•g)^3, (f/g)^3, (f o g)^3, (g o f)^3 with detailed step-by-step solutions. This problem covers key concepts in functions and their operations, suitable for Grades 11-12 students.

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