Math Problem Statement

Determine the sum and difference of each functions. f(x)=\frac{x+5}{x-3}

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

Solution

To determine the sum and difference of the functions f(x)=x+5x3f(x) = \frac{x+5}{x-3} and g(x)=x+4x2g(x) = \frac{x+4}{x-2}, we proceed as follows:

Sum of the Functions

The sum of the functions f(x)f(x) and g(x)g(x) is given by:

f(x)+g(x)=x+5x3+x+4x2f(x) + g(x) = \frac{x+5}{x-3} + \frac{x+4}{x-2}

To add these fractions, we need to find a common denominator, which is (x3)(x2)(x-3)(x-2).

f(x)+g(x)=(x+5)(x2)(x3)(x2)+(x+4)(x3)(x3)(x2)f(x) + g(x) = \frac{(x+5)(x-2)}{(x-3)(x-2)} + \frac{(x+4)(x-3)}{(x-3)(x-2)}

Now, combine the numerators:

f(x)+g(x)=(x+5)(x2)+(x+4)(x3)(x3)(x2)f(x) + g(x) = \frac{(x+5)(x-2) + (x+4)(x-3)}{(x-3)(x-2)}

Next, expand the numerators:

(x+5)(x2)=x22x+5x10=x2+3x10(x+5)(x-2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10

(x+4)(x3)=x23x+4x12=x2+x12(x+4)(x-3) = x^2 - 3x + 4x - 12 = x^2 + x - 12

Now, add the expanded numerators:

f(x)+g(x)=(x2+3x10)+(x2+x12)(x3)(x2)=2x2+4x22(x3)(x2)f(x) + g(x) = \frac{(x^2 + 3x - 10) + (x^2 + x - 12)}{(x-3)(x-2)} = \frac{2x^2 + 4x - 22}{(x-3)(x-2)}

This can be simplified further:

f(x)+g(x)=2(x2+2x11)(x3)(x2)f(x) + g(x) = \frac{2(x^2 + 2x - 11)}{(x-3)(x-2)}

Difference of the Functions

The difference of the functions f(x)f(x) and g(x)g(x) is given by:

f(x)g(x)=x+5x3x+4x2f(x) - g(x) = \frac{x+5}{x-3} - \frac{x+4}{x-2}

Again, find a common denominator:

f(x)g(x)=(x+5)(x2)(x3)(x2)(x+4)(x3)(x3)(x2)f(x) - g(x) = \frac{(x+5)(x-2)}{(x-3)(x-2)} - \frac{(x+4)(x-3)}{(x-3)(x-2)}

Now, combine the numerators:

f(x)g(x)=(x+5)(x2)(x+4)(x3)(x3)(x2)f(x) - g(x) = \frac{(x+5)(x-2) - (x+4)(x-3)}{(x-3)(x-2)}

Using the previously expanded forms:

f(x)g(x)=(x2+3x10)(x2+x12)(x3)(x2)=(x2+3x10)(x2+x12)(x3)(x2)f(x) - g(x) = \frac{(x^2 + 3x - 10) - (x^2 + x - 12)}{(x-3)(x-2)} = \frac{(x^2 + 3x - 10) - (x^2 + x - 12)}{(x-3)(x-2)}

Simplify the numerator:

f(x)g(x)=x2+3x10x2x+12(x3)(x2)=2x+2(x3)(x2)f(x) - g(x) = \frac{x^2 + 3x - 10 - x^2 - x + 12}{(x-3)(x-2)} = \frac{2x + 2}{(x-3)(x-2)}

This can be simplified further:

f(x)g(x)=2(x+1)(x3)(x2)f(x) - g(x) = \frac{2(x + 1)}{(x-3)(x-2)}

Final Answers:

  • Sum of the functions: f(x)+g(x)=2(x2+2x11)(x3)(x2)f(x) + g(x) = \frac{2(x^2 + 2x - 11)}{(x-3)(x-2)}
  • Difference of the functions: f(x)g(x)=2(x+1)(x3)(x2)f(x) - g(x) = \frac{2(x + 1)}{(x-3)(x-2)}

Would you like further details or explanations on any part of this?

Here are some related questions for further exploration:

  1. What is the product of f(x)f(x) and g(x)g(x)?
  2. How do you determine the quotient of f(x)f(x) and g(x)g(x)?
  3. Can you find the domain of f(x)+g(x)f(x) + g(x)?
  4. What is the domain of f(x)g(x)f(x) - g(x)?
  5. How would the graphs of f(x)f(x) and g(x)g(x) look like?

Tip: Always simplify your expressions as much as possible to make solving and interpreting them easier.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Fractions
Algebra
Function Operations

Formulas

-

Theorems

-

Suitable Grade Level

High School