Math Problem Statement

Let f(x) = sqrt(x), g(x) = 4x - 1, h(x) = 1/(x + 9). Find (f o h o g)(x) and its domain.

Solution

We are given the following functions:

  • f(x)=xf(x) = \sqrt{x}
  • g(x)=4x1g(x) = 4x - 1
  • h(x)=1x+9h(x) = \frac{1}{x + 9}

We are tasked with finding (fhg)(x)(f \circ h \circ g)(x), which represents f(h(g(x)))f(h(g(x))), and determining its domain.

Step 1: Evaluate h(g(x))h(g(x))

First, substitute g(x)=4x1g(x) = 4x - 1 into h(x)=1x+9h(x) = \frac{1}{x + 9}: h(g(x))=h(4x1)=1(4x1)+9=14x+8h(g(x)) = h(4x - 1) = \frac{1}{(4x - 1) + 9} = \frac{1}{4x + 8}

Step 2: Evaluate f(h(g(x)))f(h(g(x)))

Next, substitute h(g(x))=14x+8h(g(x)) = \frac{1}{4x + 8} into f(x)=xf(x) = \sqrt{x}: f(h(g(x)))=f(14x+8)=14x+8f(h(g(x))) = f\left( \frac{1}{4x + 8} \right) = \sqrt{\frac{1}{4x + 8}} So, f(h(g(x)))=14x+8f(h(g(x))) = \frac{1}{\sqrt{4x + 8}}.

Step 3: Determine the Domain

The domain is determined by the conditions that must be satisfied for the expression to be defined:

  1. For the square root function 4x+8\sqrt{4x + 8}, we require 4x+804x + 8 \geq 0, which simplifies to x2x \geq -2.
  2. The denominator 4x+84x + 8 cannot be zero because division by zero is undefined. Solving 4x+8=04x + 8 = 0, we get x=2x = -2, which must be excluded from the domain.

Thus, the domain is x>2x > -2.

Final Answer:

(fhg)(x)=14x+8(f \circ h \circ g)(x) = \frac{1}{\sqrt{4x + 8}} with the domain x>2x > -2.

Would you like more details or have any questions?


Here are 5 related questions to further explore the topic:

  1. How would the domain change if f(x)f(x) were defined as f(x)=x2f(x) = x^2 instead of x\sqrt{x}?
  2. What is the range of (fhg)(x)(f \circ h \circ g)(x)?
  3. How would you calculate the inverse of f(h(g(x)))f(h(g(x)))?
  4. How would you graph f(h(g(x)))f(h(g(x))) and indicate its domain visually?
  5. How does composition of functions affect the domain of the resulting function?

Tip: Always check both the inner and outer functions for domain restrictions when dealing with composite functions.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Domain of Functions
Radicals
Algebra

Formulas

(f o h o g)(x) = f(h(g(x)))
Square root function: sqrt(x)

Theorems

Domain Restrictions for Radical Functions
Composite Function Definition

Suitable Grade Level

Grades 10-12