Math Problem Statement

Evaluate (f/h)(1) where f(x) = -x^2 + x, g(x) = 2/(x+2), and h(x) = -2x - 1.

Solution

The given image presents three functions: f(x)f(x), g(x)g(x), and h(x)h(x), as well as a request to evaluate (fh)(1)\left( \frac{f}{h} \right)(1).

The functions are: f(x)=x2+xf(x) = -x^2 + x g(x)=2x+2g(x) = \frac{2}{x + 2} h(x)=2x1h(x) = -2x - 1

To find (fh)(1)\left( \frac{f}{h} \right)(1), we need to:

  1. Calculate f(1)f(1).
  2. Calculate h(1)h(1).
  3. Compute f(1)h(1)\frac{f(1)}{h(1)}.

Step 1: Calculate f(1)f(1)

f(1)=(1)2+1=1+1=0f(1) = -(1)^2 + 1 = -1 + 1 = 0

Step 2: Calculate h(1)h(1)

h(1)=2(1)1=21=3h(1) = -2(1) - 1 = -2 - 1 = -3

Step 3: Compute f(1)h(1)\frac{f(1)}{h(1)}

f(1)h(1)=03=0\frac{f(1)}{h(1)} = \frac{0}{-3} = 0

Thus, (fh)(1)=0\left( \frac{f}{h} \right)(1) = 0.

Would you like more details or explanations? Here are some related questions you can explore:

  1. How would the result change if we evaluated fg\frac{f}{g}(1)?
  2. What is f(x)+g(x)f(x) + g(x) evaluated at x=1x = 1?
  3. Can you find the domain of g(x)g(x)?
  4. How does the shape of f(x)f(x) look graphically?
  5. What happens to fh\frac{f}{h}(x) as xx \to \infty?

Tip: When working with function divisions, always ensure that the denominator is non-zero to avoid undefined expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Function Evaluation

Formulas

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

Theorems

Function Evaluation Theorem

Suitable Grade Level

Grades 9-11