Math Problem Statement

f(x) = 1/x g(x) = (x^2 - 2x + 3)/(x - 3) Find the domain of the composite function f(g(x))

Solution

To find the domain of the composite function f(g(x))f(g(x)), we need to determine the values of xx for which the composition f(g(x))f(g(x)) is defined.

Steps:

  1. Domain of g(x)g(x): g(x)=x22x+3x3g(x) = \frac{x^2 - 2x + 3}{x - 3} The function g(x)g(x) is undefined when the denominator is 0, i.e., when: x3=0    x=3x - 3 = 0 \implies x = 3 So, the domain of g(x)g(x) is all real numbers except x=3x = 3.

  2. Domain of f(x)f(x): f(x)=1xf(x) = \frac{1}{x} The function f(x)f(x) is undefined when x=0x = 0, so for f(g(x))f(g(x)) to be defined, g(x)0g(x) \neq 0.

  3. Solve g(x)=0g(x) = 0: To ensure that f(g(x))f(g(x)) is defined, we need to find when g(x)=0g(x) = 0: x22x+3x3=0    x22x+3=0\frac{x^2 - 2x + 3}{x - 3} = 0 \implies x^2 - 2x + 3 = 0 The quadratic equation x22x+3=0x^2 - 2x + 3 = 0 has no real solutions because its discriminant is negative: Δ=(2)24(1)(3)=412=8\Delta = (-2)^2 - 4(1)(3) = 4 - 12 = -8 Since there are no real solutions to this equation, g(x)g(x) never equals 0.

Conclusion:

The domain of f(g(x))f(g(x)) is the same as the domain of g(x)g(x), except for when g(x)=0g(x) = 0, which doesn't happen for any real xx. Therefore, the only restriction is that x3x \neq 3.

Thus, the domain of f(g(x))f(g(x)) is: x(,3)(3,)x \in (-\infty, 3) \cup (3, \infty)

Would you like a more detailed explanation of any step?


Here are 5 related questions you might find useful:

  1. How do you find the domain of the sum of two functions?
  2. What happens to the domain if the composite function has a square root?
  3. How do you find the range of a composite function?
  4. What is the difference between a composite function and a product of functions?
  5. Can the range of f(g(x))f(g(x)) affect its domain?

Tip: Always start by finding the domain of the inner function before considering the outer function when working with composite functions.

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Math Problem Analysis

Mathematical Concepts

Composite Functions
Domain of a Function
Rational Functions
Quadratic Equations

Formulas

f(x) = 1/x
g(x) = (x^2 - 2x + 3)/(x - 3)

Theorems

Domain of a composite function
Discriminant of a quadratic equation

Suitable Grade Level

Grades 9-11