Questions: The functions f and g are defined as f(x)=4x-3 and g(x)=-5x^2. a) Find the domain of f, g, f+g, f-g, fg, ff, f/g, and g/f. b) Find (f+g)(x), (f-g)(x), (fg)(x), (ff)(x), (f/g)(x), and (g/f)(x). The domain of fg is (-∞, ∞). The domain of ff is (-∞, ∞). The domain of (f/g)(x) is (-∞, 0) ∪ (0, ∞). The domain of (g/f)(x) is (-∞, 3/4) ∪ (3/4, ∞). b) (f+g)(x)=-5x^2+4x-3 (f-g)(x)=5x^2+4x-3 (fg)(x)=-5x^2(4x-3)

The functions f and g are defined as f(x)=4x-3 and g(x)=-5x^2.
a) Find the domain of f, g, f+g, f-g, fg, ff, f/g, and g/f.
b) Find (f+g)(x), (f-g)(x), (fg)(x), (ff)(x), (f/g)(x), and (g/f)(x).

The domain of fg is (-∞, ∞).
The domain of ff is (-∞, ∞).
The domain of (f/g)(x) is (-∞, 0) ∪ (0, ∞).
The domain of (g/f)(x) is (-∞, 3/4) ∪ (3/4, ∞).
b) (f+g)(x)=-5x^2+4x-3
(f-g)(x)=5x^2+4x-3
(fg)(x)=-5x^2(4x-3)

Transcript text: The functions $f$ and $g$ are defined as $f(x)=4 x-3$ and $g(x)=-5 x^{2}$. a) Find the domain of $f, g, f+g, f-g, f g, f f, \frac{f}{g}$, and $\frac{g}{f}$. b) Find $(f+g)(x),(f-g)(x)$, $(f g)(x)$, $(f f)(x),\left(\frac{f}{g}\right)(x)$, and $\left(\frac{g}{f}\right)(x)$. The domain of fg is $(-\infty, \infty)$. The domain of ff is $(-\infty, \infty)$. The domain of $\left(\frac{f}{g}\right)(x)$ is $(-\infty, 0) \cup(0, \infty)$. The domain of $\left(\frac{g}{f}\right)(x)$ is $\left(-\infty, \frac{3}{4}\right) \cup\left(\frac{3}{4}, \infty\right)$. b) $(f+g)(x)=-5 x^{2}+4 x-3$ $(f-g)(x)=5 x^{2}+4 x-3$ $(f g)(x)=-5 x^{2}(4 x-3)$

Solution

Solution Steps

Solution Approach

Domain of Functions:
- For $ f(x) = 4x - 3 $, the domain is all real numbers.
- For $ g(x) = -5x^2 $, the domain is all real numbers.
- For $ f+g $, $ f-g $, $ fg $, and $ ff $, the domain is the intersection of the domains of $ f $ and $ g $, which is all real numbers.
- For $ \frac{f}{g} $ and $ \frac{g}{f} $, the domain excludes points where the denominator is zero.
Function Operations:
- $ (f+g)(x) = f(x) + g(x) $
- $ (f-g)(x) = f(x) - g(x) $
- $ (fg)(x) = f(x) \cdot g(x) $
- $ (ff)(x) = f(f(x)) $
- $ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} $
- $ \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} $

Step 1: Domain of Functions

The functions are defined as follows:

$ f(x) = 4x - 3 $ has a domain of $ (-\infty, \infty) $.
$ g(x) = -5x^2 $ also has a domain of $ (-\infty, \infty) $.

For the combined functions:

The domain of $ f + g $ is $ (-\infty, \infty) $.
The domain of $ f - g $ is $ (-\infty, \infty) $.
The domain of $ fg $ is $ (-\infty, \infty) $.
The domain of $ ff $ is $ (-\infty, \infty) $.
The domain of $ \frac{f}{g} $ is $ (-\infty, 0) \cup (0, \infty) $ (excluding $ x = 0 $).
The domain of $ \frac{g}{f} $ is $ (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty) $ (excluding $ x = \frac{3}{4} $).

Step 2: Function Operations

Calculating the combined functions:

$ (f + g)(x) = -5x^2 + 4x - 3 $
$ (f - g)(x) = 5x^2 + 4x - 3 $
$ (fg)(x) = -5x^2(4x - 3) $
$ (ff)(x) = 16x - 15 $
$ \left(\frac{f}{g}\right)(x) = -\frac{4x - 3}{5x^2} $
$ \left(\frac{g}{f}\right)(x) = -\frac{5x^2}{4x - 3} $

Final Answer

Domain of $ f $: $ \boxed{(-\infty, \infty)} $
Domain of $ g $: $ \boxed{(-\infty, \infty)} $
Domain of $ f + g $: $ \boxed{(-\infty, \infty)} $
Domain of $ f - g $: $ \boxed{(-\infty, \infty)} $
Domain of $ fg $: $ \boxed{(-\infty, \infty)} $
Domain of $ ff $: $ \boxed{(-\infty, \infty)} $
Domain of $ \frac{f}{g} $: $ \boxed{(-\infty, 0) \cup (0, \infty)} $
Domain of $ \frac{g}{f} $: $ \boxed{(-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty)} $
$ (f + g)(x) = \boxed{-5x^2 + 4x - 3} $
$ (f - g)(x) = \boxed{5x^2 + 4x - 3} $
$ (fg)(x) = \boxed{-5x^2(4x - 3)} $
$ (ff)(x) = \boxed{16x - 15} $
$ \left(\frac{f}{g}\right)(x) = \boxed{-\frac{4x - 3}{5x^2}} $
$ \left(\frac{g}{f}\right)(x) = \boxed{-\frac{5x^2}{4x - 3}} $

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