Math Problem Statement

f(x)=x^2-5x+6 g(x)=x-3 find (f/g)

Solution

To find (fg)(x)\left(\frac{f}{g}\right)(x), we divide f(x)f(x) by g(x)g(x). The given functions are:

  • f(x)=x25x+6f(x) = x^2 - 5x + 6
  • g(x)=x3g(x) = x - 3

Now, we compute:

(fg)(x)=f(x)g(x)=x25x+6x3\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x + 6}{x - 3}

We can simplify this by factoring the numerator x25x+6x^2 - 5x + 6:

x25x+6=(x2)(x3)x^2 - 5x + 6 = (x - 2)(x - 3)

Thus, we get:

(fg)(x)=(x2)(x3)x3\left(\frac{f}{g}\right)(x) = \frac{(x - 2)(x - 3)}{x - 3}

For x3x \neq 3, the (x3)(x - 3) terms cancel out, and we are left with:

(fg)(x)=x2forx3\left(\frac{f}{g}\right)(x) = x - 2 \quad \text{for} \, x \neq 3

So, the simplified function is:

(fg)(x)=x2withx3\left(\frac{f}{g}\right)(x) = x - 2 \quad \text{with} \, x \neq 3

Would you like further details or explanations?

Here are 5 related questions to expand on this:

  1. What happens to the function when x=3x = 3?
  2. How do you factor other quadratic expressions like x25x+6x^2 - 5x + 6?
  3. What is the domain of the function (fg)(x)\left(\frac{f}{g}\right)(x)?
  4. Can you graph the function (fg)(x)\left(\frac{f}{g}\right)(x) and explain its key features?
  5. How do you apply division of functions to real-life scenarios?

Tip: Always check for undefined points (like x=3x = 3 here) when dividing functions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Factoring Quadratic Expressions
Domain of Functions

Formulas

f(x) = x^2 - 5x + 6
g(x) = x - 3
(f/g)(x) = f(x)/g(x)

Theorems

Factoring Theorem
Simplification of Rational Expressions

Suitable Grade Level

Grades 9-10