Math Problem Statement

Determine the domains of the functions f(x) = 1/x and g(x) = 1/(5x), as well as the domain of the composition of these functions f ∘ g.

Solution

The problem in the image is asking to determine the domains of two functions f(x)=1xf(x) = \frac{1}{x} and g(x)=15xg(x) = \frac{1}{5x}, as well as the domain of the composition of these functions (fg)(x)(f \circ g)(x).

Solution:

(a) Domain of f(x)=1xf(x) = \frac{1}{x}

The domain of a function is the set of all input values xx for which the function is defined. For f(x)=1xf(x) = \frac{1}{x}, the function is undefined when the denominator is zero. Therefore: x0x \neq 0 The domain of f(x)f(x) is all real numbers except zero, which in interval notation is: (,0)(0,)(-\infty, 0) \cup (0, \infty)

(b) Domain of g(x)=15xg(x) = \frac{1}{5x}

Similarly, for g(x)=15xg(x) = \frac{1}{5x}, the denominator is 5x5x, and the function is undefined when 5x=05x = 0, or equivalently when x=0x = 0. Thus, the domain of g(x)g(x) is: (,0)(0,)(-\infty, 0) \cup (0, \infty)

(c) Domain of (fg)(x)(f \circ g)(x)

The composition (fg)(x)=f(g(x))(f \circ g)(x) = f(g(x)) means applying ff to the output of g(x)g(x). First, g(x)=15xg(x) = \frac{1}{5x}, so: f(g(x))=f(15x)=115x=5xf(g(x)) = f\left(\frac{1}{5x}\right) = \frac{1}{\frac{1}{5x}} = 5x Now, f(g(x))=5xf(g(x)) = 5x is a linear function, which is defined for all real numbers except when x=0x = 0, since g(x)g(x) is undefined at x=0x = 0. Therefore, the domain of fgf \circ g is also: (,0)(0,)(-\infty, 0) \cup (0, \infty)

Summary:

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

Would you like further clarification or details on any part of the solution?

Here are 5 related questions to deepen your understanding:

  1. How do you determine the domain of a function involving fractions?
  2. What happens if you compose two functions with different domains?
  3. How do you graph a function like f(x)=1xf(x) = \frac{1}{x}?
  4. What is the general rule for determining the domain of a composite function?
  5. Can there be other restrictions on the domain besides the denominator being zero?

Tip: Always check for points where a function is undefined, such as divisions by zero or negative values under a square root, when finding the domain.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Domain of a function
Composite functions
Algebra

Formulas

f(x) = 1/x
g(x) = 1/(5x)
(f ∘ g)(x) = f(g(x))

Theorems

The domain of a rational function is all real numbers except where the denominator equals zero.

Suitable Grade Level

Grades 10-12