Math Problem Statement

For the functions f(x) = 2/(x+3) and g(x) = 7/x, find the composition f ∘ g and simplify your answer as much as possible. Write the domain using interval notation.

Solution

To solve this problem, we are asked to find the composition of two functions ff and gg, given by:

f(x)=2x+3andg(x)=7xf(x) = \frac{2}{x+3} \quad \text{and} \quad g(x) = \frac{7}{x}

We need to find (fg)(x)(f \circ g)(x), which is the composition of ff and gg. This means that we will substitute g(x)g(x) into f(x)f(x).

Step-by-Step Solution:

  1. Substitute g(x)=7xg(x) = \frac{7}{x} into f(x)f(x):

    (fg)(x)=f(g(x))=f(7x)(f \circ g)(x) = f(g(x)) = f\left(\frac{7}{x}\right)

  2. Substitute 7x\frac{7}{x} for xx in f(x)f(x):

    f(7x)=27x+3f\left(\frac{7}{x}\right) = \frac{2}{\frac{7}{x} + 3}

  3. Simplify the expression inside the denominator:

    7x+3=7x+3xx=7+3xx\frac{7}{x} + 3 = \frac{7}{x} + \frac{3x}{x} = \frac{7 + 3x}{x}

    So, the expression becomes:

    f(7x)=27+3xx=2x7+3xf\left(\frac{7}{x}\right) = \frac{2}{\frac{7 + 3x}{x}} = 2 \cdot \frac{x}{7 + 3x}

  4. Final simplified result:

    (fg)(x)=2x7+3x(f \circ g)(x) = \frac{2x}{7 + 3x}

Domain of fgf \circ g:

To find the domain, we need to determine where the expression is defined. There are two cases to consider:

  1. From g(x)g(x): Since g(x)=7xg(x) = \frac{7}{x}, the function is undefined when x=0x = 0 (division by zero).

  2. From f(g(x))f(g(x)): The expression 2x7+3x\frac{2x}{7 + 3x} is undefined when the denominator 7+3x=07 + 3x = 0, i.e., when x=73x = -\frac{7}{3}.

Thus, the domain is all real numbers except x=0x = 0 and x=73x = -\frac{7}{3}.

Domain in interval notation:

The domain is: (,73)(73,0)(0,)(-\infty, -\frac{7}{3}) \cup (-\frac{7}{3}, 0) \cup (0, \infty)

Final Answer:

(fg)(x)=2x7+3x(f \circ g)(x) = \frac{2x}{7 + 3x} Domain of fg:(,73)(73,0)(0,)\text{Domain of } f \circ g: (-\infty, -\frac{7}{3}) \cup (-\frac{7}{3}, 0) \cup (0, \infty)


Let me know if you'd like further details or clarifications! Here are some related questions you might find interesting:

  1. How do you determine the domain of a composed function in general?
  2. Can you explain the difference between domain and range in function composition?
  3. What happens if we reverse the composition and find (gf)(x)(g \circ f)(x)?
  4. How would the range of fgf \circ g be determined?
  5. What role do asymptotes play in determining the behavior of the function fgf \circ g?

Tip: Always simplify the inner function first before composing with the outer function. This often helps in identifying any restrictions on the domain early on.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra
Domain and Range

Formulas

f(x) = 2 / (x + 3)
g(x) = 7 / x
f ∘ g(x) = f(g(x)) = 2 / (g(x) + 3)

Theorems

Domain of Rational Functions
Composition of Functions

Suitable Grade Level

Grades 10-12