Question

For the given functions fand g. find the following. For parts \((a)-(d),\) also find the domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f \cdot g)(x)\) (d) \(\left(\frac{f}{g}\right)(x)\) (e) \((f+g)\) (3) (f) \((f-g)\) (4) (g) \((f \cdot g)\) ( 2 ) \((h)\left(\frac{f}{g}\right)(1)\) \(f(x)=\frac{2 x+3}{3 x-2}, \quad g(x)=\frac{4 x}{3 x-2}\)

Short answer

The domains for (a)-(d) exclude x = 2/3, with additional exclusion of x = 0 for (d). Results: (a) 3, (b) -1/2, (c) 3.5, (d) 1.25.

Step-by-step solution

- Define the functions

Given: \[ f(x) = \frac{2x + 3}{3x - 2} \] \[ g(x) = \frac{4x}{3x - 2} \]

- Find the domain of f(x) and g(x)

The domain of a function is all values of x for which the function is defined. Both functions have a denominator of \(3x - 2\). So, set the denominator not equal to zero and solve for x: \[ 3x - 2 eq 0 \] \[ x eq \frac{2}{3} \] Thus, the domain for both \(f(x)\) and \(g(x)\) is all real numbers except \(\frac{2}{3}\).

- (a) Find (f+g)(x) and its domain

Calculate \((f + g)(x)\): \[ (f + g)(x) = \frac{2x + 3}{3x - 2} + \frac{4x}{3x - 2} \] Since the denominators are the same, combine the numerators: \[ (f + g)(x) = \frac{(2x + 3) + 4x}{3x - 2} = \frac{6x + 3}{3x - 2} \] The domain is all real numbers except \(\frac{2}{3}\).

- (b) Find (f-g)(x) and its domain

Calculate \((f - g)(x)\): \[ (f - g)(x) = \frac{2x + 3}{3x - 2} - \frac{4x}{3x - 2} \] Since the denominators are the same, combine the numerators: \[ (f - g)(x) = \frac{(2x + 3) - 4x}{3x - 2} = \frac{-2x + 3}{3x - 2} \] The domain is all real numbers except \(\frac{2}{3}\).

- (c) Find (f * g)(x) and its domain

Calculate \((f \cdot g)(x)\): \[ (f \cdot g)(x) = \frac{2x + 3}{3x - 2} \cdot \frac{4x}{3x - 2} = \frac{(2x + 3) \cdot 4x}{(3x - 2)^2} = \frac{8x^2 + 12x}{9x^2 - 12x + 4} \] The domain is all real numbers except \(\frac{2}{3}\).

- (d) Find (f/g)(x) and its domain

Calculate \(\left(\frac{f}{g}\right)(x)\): \[ \left(\frac{f}{g}\right)(x) = \frac{\frac{2x + 3}{3x - 2}}{\frac{4x}{3x - 2}} = \frac{2x + 3}{4x} = \frac{2x + 3}{4x} \] Simplifying the expression: \[ \left(\frac{f}{g}\right)(x) = \frac{2x + 3}{4x} \] The domain is all real numbers except \(0\) and \(\frac{2}{3}\).

- (e) Find (f+g)(3)

Substitute x = 3 into \((f + g)(x)\): \[ (f + g)(3) = \frac{6(3) + 3}{3(3) - 2} = \frac{18 + 3}{9 - 2} = \frac{21}{7} = 3 \]

- (f) Find (f-g)(4)

Substitute x = 4 into \((f - g)(x)\): \[ (f - g)(4) = \frac{-2(4) + 3}{3(4) - 2} = \frac{-8 + 3}{12 - 2} = \frac{-5}{10} = -\frac{1}{2} \]

- (g) Find (f * g)(2)

Substitute x = 2 into \((f \cdot g)(x)\): \[ (f \cdot g)(2) = \frac{8(2^2) + 12(2)}{9(2^2) - 12(2) + 4} = \frac{8(4) + 24}{9(4) - 24 + 4} = \frac{32 + 24}{36 - 24 + 4} = \frac{56}{16} = 3.5 \]

- (h) Find (f/g)(1)

Substitute x = 1 into \(\left(\frac{f}{g}\right)(x)\): \[ \left(\frac{f}{g}\right)(1) = \frac{2(1) + 3}{4(1)} = \frac{5}{4} = 1.25 \]

Key concepts

Functions

Functions are mathematical entities that describe a relationship between inputs (often denoted as x) and outputs (often represented as f(x) or g(x)). Each input must map to exactly one output. Functions can be represented in various forms, including equations, tables, and graphs.
For example, consider the functions given:

• f(x) = \( \frac{2x + 3}{3x - 2} \)
• g(x) = \( \frac{4x}{3x - 2} \)

These functions use algebraic expressions to define outputs based on inputs. Different operations like addition, subtraction, multiplication, and division of functions help to analyze various properties and behaviors of these relationships.

Domain

The domain of a function is the set of all possible input values (x) for which the function is defined. When dealing with rational functions (fractions), the domain excludes any x-values that make the denominator equal to zero, as this would result in an undefined expression.
For the functions:

• f(x) = \( \frac{2x + 3}{3x - 2} \)
• g(x) = \( \frac{4x}{3x - 2} \)

The domain for both functions is all real numbers except where the denominator equals zero. Solving \( 3x - 2 eq 0 \) yields \( x eq \frac{2}{3} \). Thus, the domain is all real numbers except \( \frac{2}{3} \).

Function Addition

Function addition involves adding the outputs of two functions for any input x. To find \( (f + g)(x) \), simply add the algebraic expressions for f(x) and g(x). Since the denominators are the same, you can directly combine the numerators:
\[ (f + g)(x) = \frac{2x + 3}{3x - 2} + \frac{4x}{3x - 2} = \frac{(2x + 3) + 4x}{3x - 2} = \frac{6x + 3}{3x - 2} \]
The domain is the intersection of the domains of f and g, meaning all real numbers except \( \frac{2}{3} \).
Also, evaluating this at x=3 gives:
\[ (f + g)(3) = \frac{6(3) + 3}{3(3) - 2} = \frac{21}{7} = 3 \]

Function Subtraction

Function subtraction involves subtracting the outputs of one function from another for any input x. To find \( (f - g)(x) \), subtract the algebraic expressions for f(x) and g(x):
\[ (f - g)(x) = \frac{2x + 3}{3x - 2} - \frac{4x}{3x - 2} = \frac{(2x + 3) - 4x}{3x - 2} = \frac{-2x + 3}{3x - 2} \]
The domain, like in the addition, is all real numbers except \( \frac{2}{3} \).
Also, evaluating this at x=4 gives:
\[ (f - g)(4) = \frac{-2(4) + 3}{3(4) - 2} = \frac{-5}{10} = -\frac{1}{2} \]

Function Multiplication

Function multiplication involves multiplying the outputs of two functions for any input x. To find \( (f \cdot g)(x) \), multiply the algebraic expressions for f(x) and g(x):
\[ (f \cdot g)(x) = \frac{2x + 3}{3x - 2} \cdot \frac{4x}{3x - 2} = \frac{(2x + 3) \cdot 4x}{(3x - 2)^2} = \frac{8x^2 + 12x}{9x^2 - 12x + 4} \]
The domain is all real numbers except \( \frac{2}{3} \).
Also, evaluating this at x=2 gives:
\[ (f \cdot g)(2) = \frac{8(4) + 24}{36 - 24 + 4} = \frac{56}{16} = 3.5 \]

Function Division

Function division involves dividing the output of one function by another for any input x. To find \( \left(\frac{f}{g}\right)(x) \), divide the algebraic expressions for f(x) and g(x):
\[ \left(\frac{f}{g}\right)(x) = \frac{\frac{2x + 3}{3x - 2}}{\frac{4x}{3x - 2}} = \frac{2x + 3}{4x} = \frac{2x + 3}{4x} \]
The domain is all real numbers except where x makes the denominator zero, which in this case is \( x eq 0 \) and \( x eq \frac{2}{3} \).
Also, evaluating this at x=1 gives:
\[ \left(\frac{f}{g}\right)(1) = \frac{2(1) + 3}{4(1)} = \frac{5}{4} = 1.25 \]