Answer :
Let's address this question step by step.
### Part (a): Finding the Domain
1. Domain of \( f(x) \):
- \( f(x) = 4x - 1 \)
- This is a linear function, and linear functions are defined for all real numbers.
- Domain of \( f \): \((-\infty, \infty)\)
2. Domain of \( g(x) \):
- \( g(x) = -5x^2 \)
- This is a quadratic function, and quadratic functions are defined for all real numbers.
- Domain of \( g \): \((-\infty, \infty)\)
3. Domain of \( f+g \):
- \( f+g(x) = f(x) + g(x) = 4x - 1 - 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( f+g \): \((-\infty, \infty)\)
4. Domain of \( f-g \):
- \( f-g(x) = f(x) - g(x) = 4x - 1 + 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( f-g \): \((-\infty, \infty)\)
5. Domain of \( fg \):
- \( fg(x) = f(x) \cdot g(x) = (4x - 1)(-5x^2) = -20x^3 + 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( fg \): \((-\infty, \infty)\)
6. Domain of \( ff \):
- We interpret \( ff(x) \) as \( f(x) \cdot f(x) = (4x - 1)(4x - 1) = 16x^2 - 8x + 1 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( ff \): \((-\infty, \infty)\)
7. Domain of \( \frac{f}{g} \):
- \( \left(\frac{f}{g}\right)(x) = \frac{4x - 1}{-5x^2} \)
- The denominator \(-5x^2\) must not equal zero.
- \( x \neq 0 \)
- Domain of \( \frac{f}{g} \): \((-\infty, 0) \cup (0, \infty)\)
8. Domain of \( \frac{g}{f} \):
- \( \left(\frac{g}{f}\right)(x) = \frac{-5x^2}{4x - 1} \)
- The denominator \( 4x - 1 \) must not equal zero.
- \( 4x - 1 \neq 0 \)
- \( x \neq \frac{1}{4} \)
- Domain of \( \frac{g}{f} \): \( (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty) \)
So, the detailed domains are:
- \( f \): \( (-\infty, \infty) \)
- \( g \): \( (-\infty, \infty) \)
- \( f+g \): \( (-\infty, \infty) \)
- \( f-g \): \( (-\infty, \infty) \)
- \( fg \): \( (-\infty, \infty) \)
- \( ff \): \( (-\infty, \infty) \)
- \( \frac{f}{g} \): \( (-\infty, \infty) \), excluding \( x = 0 \)
- \( \frac{g}{f} \): \( (-\infty, \infty) \), excluding \( x = \frac{1}{4} \)
### Part (b): Finding the Composed Functions
Given \( f(x) = 4x - 1 \) and \( g(x) = -5x^2 \):
1. \( (f+g)(x) \):
- \( (f+g)(x) = f(x) + g(x) = 4x - 1 - 5x^2 \)
- So, \((f+g)(x) = -5x^2 + 4x - 1\)
2. \( (f-g)(x) \):
- \( (f-g)(x) = f(x) - g(x) = 4x - 1 - (-5x^2) \)
- So, \((f-g)(x) = 5x^2 + 4x - 1\)
3. \( (fg)(x) \):
- \( (fg)(x) = f(x) \cdot g(x) = (4x - 1)(-5x^2) \)
- So, \( (fg)(x) = -20x^3 + 5x^2 \)
4. \( (ff)(x) \):
- \( (ff)(x) = f(x) \cdot f(x) = (4x - 1)^2 \)
- So, \( (ff)(x) = 16x^2 - 8x + 1\)
5. \( \left(\frac{f}{g}\right)(x) \):
- \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x - 1}{-5x^2} \)
- This is valid for \( x \neq 0\)
6. \( \left(\frac{g}{f}\right)(x) \):
- \( \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{-5x^2}{4x - 1} \)
- This is valid for \( x \neq \frac{1}{4} \)
### Evaluation at \( x = 1 \)
Let's evaluate each function at \( x = 1 \):
- \( f(1) = 4 \cdot 1 - 1 = 3 \)
- \( g(1) = -5 \cdot 1^2 = -5 \)
- \( ff(1) = (4 \cdot 1 - 1)^2 = 3^2 = 9 \)
- \( (f+g)(1) = -5 \cdot 1^2 + 4 \cdot 1 - 1 = -5 + 4 - 1 = -2 \)
- \( (f-g)(1) = 5 \cdot 1^2 + 4 \cdot 1 - 1 = 5 + 4 - 1 = 8 \)
- \( (fg)(1) = (4 \cdot 1 - 1)(-5 \cdot 1^2) = 3 \cdot -5 = -15 \)
- \( \left(\frac{f}{g}\right)(1) = \frac{4 \cdot 1 - 1}{-5 \cdot 1^2} = \frac{3}{-5} = -0.6 \)
- \( \left(\frac{g}{f}\right)(1) = \frac{-5 \cdot 1^2}{4 \cdot 1 - 1} = \frac{-5}{3} \approx -1.66667 \)
### Summary
- The domains:
- \( f \): \( (-\infty, \infty) \)
- \( g \): \( (-\infty, \infty) \)
- \( f+g \): \( (-\infty, \infty) \)
- \( f-g \): \( (-\infty, \infty) \)
- \( fg \): \( (-\infty, \infty) \)
- \( ff \): \( (-\infty, \infty) \)
- \( \frac{f}{g} \): \( (-\infty, \infty) \), excluding \( x = 0 \)
- \( \frac{g}{f} \): \( (-\infty, \infty) \), excluding \( x = \frac{1}{4} \)
- The values at \( x = 1 \):
- \( f(1) = 3 \)
- \( g(1) = -5 \)
- \( ff(1) = 9 \)
- \( (f+g)(1) = -2 \)
- \( (f-g)(1) = 8 \)
- \( (fg)(1) = -15 \)
- \( \left(\frac{f}{g}\right)(1) = -0.6 \)
- [tex]\( \left(\frac{g}{f}\right)(1) \approx -1.66667 \)[/tex]
### Part (a): Finding the Domain
1. Domain of \( f(x) \):
- \( f(x) = 4x - 1 \)
- This is a linear function, and linear functions are defined for all real numbers.
- Domain of \( f \): \((-\infty, \infty)\)
2. Domain of \( g(x) \):
- \( g(x) = -5x^2 \)
- This is a quadratic function, and quadratic functions are defined for all real numbers.
- Domain of \( g \): \((-\infty, \infty)\)
3. Domain of \( f+g \):
- \( f+g(x) = f(x) + g(x) = 4x - 1 - 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( f+g \): \((-\infty, \infty)\)
4. Domain of \( f-g \):
- \( f-g(x) = f(x) - g(x) = 4x - 1 + 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( f-g \): \((-\infty, \infty)\)
5. Domain of \( fg \):
- \( fg(x) = f(x) \cdot g(x) = (4x - 1)(-5x^2) = -20x^3 + 5x^2 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( fg \): \((-\infty, \infty)\)
6. Domain of \( ff \):
- We interpret \( ff(x) \) as \( f(x) \cdot f(x) = (4x - 1)(4x - 1) = 16x^2 - 8x + 1 \)
- This is a polynomial, and polynomials are defined for all real numbers.
- Domain of \( ff \): \((-\infty, \infty)\)
7. Domain of \( \frac{f}{g} \):
- \( \left(\frac{f}{g}\right)(x) = \frac{4x - 1}{-5x^2} \)
- The denominator \(-5x^2\) must not equal zero.
- \( x \neq 0 \)
- Domain of \( \frac{f}{g} \): \((-\infty, 0) \cup (0, \infty)\)
8. Domain of \( \frac{g}{f} \):
- \( \left(\frac{g}{f}\right)(x) = \frac{-5x^2}{4x - 1} \)
- The denominator \( 4x - 1 \) must not equal zero.
- \( 4x - 1 \neq 0 \)
- \( x \neq \frac{1}{4} \)
- Domain of \( \frac{g}{f} \): \( (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty) \)
So, the detailed domains are:
- \( f \): \( (-\infty, \infty) \)
- \( g \): \( (-\infty, \infty) \)
- \( f+g \): \( (-\infty, \infty) \)
- \( f-g \): \( (-\infty, \infty) \)
- \( fg \): \( (-\infty, \infty) \)
- \( ff \): \( (-\infty, \infty) \)
- \( \frac{f}{g} \): \( (-\infty, \infty) \), excluding \( x = 0 \)
- \( \frac{g}{f} \): \( (-\infty, \infty) \), excluding \( x = \frac{1}{4} \)
### Part (b): Finding the Composed Functions
Given \( f(x) = 4x - 1 \) and \( g(x) = -5x^2 \):
1. \( (f+g)(x) \):
- \( (f+g)(x) = f(x) + g(x) = 4x - 1 - 5x^2 \)
- So, \((f+g)(x) = -5x^2 + 4x - 1\)
2. \( (f-g)(x) \):
- \( (f-g)(x) = f(x) - g(x) = 4x - 1 - (-5x^2) \)
- So, \((f-g)(x) = 5x^2 + 4x - 1\)
3. \( (fg)(x) \):
- \( (fg)(x) = f(x) \cdot g(x) = (4x - 1)(-5x^2) \)
- So, \( (fg)(x) = -20x^3 + 5x^2 \)
4. \( (ff)(x) \):
- \( (ff)(x) = f(x) \cdot f(x) = (4x - 1)^2 \)
- So, \( (ff)(x) = 16x^2 - 8x + 1\)
5. \( \left(\frac{f}{g}\right)(x) \):
- \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x - 1}{-5x^2} \)
- This is valid for \( x \neq 0\)
6. \( \left(\frac{g}{f}\right)(x) \):
- \( \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{-5x^2}{4x - 1} \)
- This is valid for \( x \neq \frac{1}{4} \)
### Evaluation at \( x = 1 \)
Let's evaluate each function at \( x = 1 \):
- \( f(1) = 4 \cdot 1 - 1 = 3 \)
- \( g(1) = -5 \cdot 1^2 = -5 \)
- \( ff(1) = (4 \cdot 1 - 1)^2 = 3^2 = 9 \)
- \( (f+g)(1) = -5 \cdot 1^2 + 4 \cdot 1 - 1 = -5 + 4 - 1 = -2 \)
- \( (f-g)(1) = 5 \cdot 1^2 + 4 \cdot 1 - 1 = 5 + 4 - 1 = 8 \)
- \( (fg)(1) = (4 \cdot 1 - 1)(-5 \cdot 1^2) = 3 \cdot -5 = -15 \)
- \( \left(\frac{f}{g}\right)(1) = \frac{4 \cdot 1 - 1}{-5 \cdot 1^2} = \frac{3}{-5} = -0.6 \)
- \( \left(\frac{g}{f}\right)(1) = \frac{-5 \cdot 1^2}{4 \cdot 1 - 1} = \frac{-5}{3} \approx -1.66667 \)
### Summary
- The domains:
- \( f \): \( (-\infty, \infty) \)
- \( g \): \( (-\infty, \infty) \)
- \( f+g \): \( (-\infty, \infty) \)
- \( f-g \): \( (-\infty, \infty) \)
- \( fg \): \( (-\infty, \infty) \)
- \( ff \): \( (-\infty, \infty) \)
- \( \frac{f}{g} \): \( (-\infty, \infty) \), excluding \( x = 0 \)
- \( \frac{g}{f} \): \( (-\infty, \infty) \), excluding \( x = \frac{1}{4} \)
- The values at \( x = 1 \):
- \( f(1) = 3 \)
- \( g(1) = -5 \)
- \( ff(1) = 9 \)
- \( (f+g)(1) = -2 \)
- \( (f-g)(1) = 8 \)
- \( (fg)(1) = -15 \)
- \( \left(\frac{f}{g}\right)(1) = -0.6 \)
- [tex]\( \left(\frac{g}{f}\right)(1) \approx -1.66667 \)[/tex]
