Math Problem Statement

Determine the domains of the following functions: f(x) = x/(x^2 - 1) and g(x) = x + 1. Find the domains of f, g, and the composition f(g).

Solution

The problem involves finding the domains of the given functions. Let's go through each part of the problem.

Functions Given

  1. f(x)=xx21f(x) = \frac{x}{x^2 - 1}
  2. g(x)=x+1g(x) = x + 1

(a) Domain of ff

For f(x)=xx21f(x) = \frac{x}{x^2 - 1}, the function is undefined where the denominator is zero. Therefore: x21=0    x2=1    x=±1x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1 The domain of ff is all real numbers except x=1x = 1 and x=1x = -1. In interval notation: Domain of f=(,1)(1,1)(1,)\text{Domain of } f = (-\infty, -1) \cup (-1, 1) \cup (1, \infty)

(b) Domain of gg

For g(x)=x+1g(x) = x + 1, this is a linear function, which is defined for all real numbers. So: Domain of g=(,)\text{Domain of } g = (-\infty, \infty)

(c) Domain of fgf \circ g (Composition of ff and gg)

To find the domain of fgf \circ g, we need to evaluate f(g(x))f(g(x)). The function f(g(x))=g(x)(g(x))21=x+1(x+1)21f(g(x)) = \frac{g(x)}{(g(x))^2 - 1} = \frac{x + 1}{(x + 1)^2 - 1}.

We need to find when the denominator is zero: (x+1)21=0(x + 1)^2 - 1 = 0 (x+11)(x+1+1)=0(x + 1 - 1)(x + 1 + 1) = 0 x(x+2)=0x(x + 2) = 0 This results in x=0x = 0 or x=2x = -2.

So, the domain of fgf \circ g is all real numbers except x=0x = 0 and x=2x = -2. In interval notation: Domain of fg=(,2)(2,0)(0,)\text{Domain of } f \circ g = (-\infty, -2) \cup (-2, 0) \cup (0, \infty)

Summary of Answers

  • (a) Domain of ff: (,1)(1,1)(1,)(-\infty, -1) \cup (-1, 1) \cup (1, \infty)
  • (b) Domain of gg: (,)(-\infty, \infty)
  • (c) Domain of fgf \circ g: (,2)(2,0)(0,)(-\infty, -2) \cup (-2, 0) \cup (0, \infty)

Would you like further details or have any questions?

Here are 5 related questions that expand on the above:

  1. What is the range of f(x)f(x)?
  2. How would the domain change if g(x)=x2+1g(x) = x^2 + 1?
  3. Can you find the domain of gfg \circ f instead of fgf \circ g?
  4. How does adding a constant to the numerator of f(x)f(x) affect its domain?
  5. If g(x)g(x) were a trigonometric function, how would you approach finding the domain of fgf \circ g?

Tip: Always start by finding points where the denominator is zero when working with rational functions to determine the domain.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Composition of Functions

Formulas

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

Theorems

To find the domain of a rational function, exclude values that make the denominator zero.
For composition of functions, exclude values that make the inner or outer function undefined.

Suitable Grade Level

Grades 10-12