Math Problem Statement

what is the domain of the composite function f little circle g f(x) =6x;g(x)=1x4\sqrt{6-x};g\left(x\right)=\frac{1}{x-4}

Solution

To find the domain of the composite function (fg)(x)=f(g(x))(f \circ g)(x) = f(g(x)), where:

  • f(x)=6xf(x) = \sqrt{6 - x}
  • g(x)=1x4g(x) = \frac{1}{x - 4}

we need to follow these steps:

1. Find the domain of g(x)g(x).

For g(x)=1x4g(x) = \frac{1}{x - 4}, the denominator cannot be zero, so: x40x4x - 4 \neq 0 \quad \Rightarrow \quad x \neq 4 Thus, the domain of g(x)g(x) is all real numbers except x=4x = 4, or: Domain of g(x)=(,4)(4,)\text{Domain of } g(x) = (-\infty, 4) \cup (4, \infty)

2. Find the domain of f(g(x))f(g(x)).

Now, we apply g(x)g(x) inside f(x)f(x), so: f(g(x))=f(1x4)=61x4f(g(x)) = f\left( \frac{1}{x - 4} \right) = \sqrt{6 - \frac{1}{x - 4}} For the square root to be defined, the expression inside the square root must be non-negative: 61x406 - \frac{1}{x - 4} \geq 0 Solve this inequality: 61x46 \geq \frac{1}{x - 4} Multiply both sides by x4x - 4, keeping in mind that the direction of the inequality may change depending on whether x4x - 4 is positive or negative:

  • Case 1: x4>0x - 4 > 0 (i.e., x>4x > 4)

    In this case, multiply as usual: 6(x4)16(x - 4) \geq 1 Simplifying: 6x2416x25x2566x - 24 \geq 1 \quad \Rightarrow \quad 6x \geq 25 \quad \Rightarrow \quad x \geq \frac{25}{6}

  • Case 2: x4<0x - 4 < 0 (i.e., x<4x < 4)

    Here, the inequality flips when multiplying: 6(x4)16(x - 4) \leq 1 Simplifying: 6x2416x25x2566x - 24 \leq 1 \quad \Rightarrow \quad 6x \leq 25 \quad \Rightarrow \quad x \leq \frac{25}{6}

3. Combine the results.

From Case 1, we know x256x \geq \frac{25}{6}, and from Case 2, x256x \leq \frac{25}{6}. Therefore, the solution is that x=256x = \frac{25}{6}, and we must also exclude x=4x = 4.

Conclusion:

The domain of the composite function (fg)(x)(f \circ g)(x) is {256}\left\{ \frac{25}{6} \right\}.

Would you like more details or have any further questions?

Related questions:

  1. How do you solve compound inequalities involving fractions?
  2. What is the domain of the square root function in general?
  3. How can you graphically represent composite functions?
  4. How do you handle domains when composing two functions?
  5. What are some common mistakes when finding domains of composite functions?

Tip:

Always remember to check both the inner and outer functions' domains when dealing with composite functions.

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Math Problem Analysis

Mathematical Concepts

Domain of Functions
Composite Functions
Inequalities
Square Roots
Rational Functions

Formulas

Domain restrictions for square root functions: expression inside the square root must be non-negative
Domain restrictions for rational functions: denominator cannot be zero

Theorems

Properties of Composite Functions
Inequality Multiplication Rule

Suitable Grade Level

Grades 10-12