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Chapter 3: Problem 79

Let \(f(x)=-x^{3}+2 x-2\) and \(g(x)=\frac{2-x}{9+x}\) and find each value. $$ g(2) $$

Short Answer

Expert verified
The value of \( g(2) \) is 0.

Step by step solution

01

- Substitute the Value into the Function

To find the value of \( g(2) \), substitute \( x = 2 \) into the function \( g(x) = \frac{2-x}{9+x} \). This gives us the expression \( g(2) = \frac{2-2}{9+2} \).
02

- Simplify the Expression

Evaluate the numerator and denominator separately. The numerator is \( 2-2 = 0 \) and the denominator is \( 9+2 = 11 \). Thus, our expression is \( \frac{0}{11} \).
03

- Compute the Final Value

Since \( \frac{0}{11} = 0 \), the value of \( g(2) \) is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a fundamental technique used in function evaluation. It involves replacing a variable in a function with a specific value to simplify and solve the equation. In our exercise, we're asked to find the value of the function \(g(x) = \frac{2-x}{9+x}\) when \(x = 2\). To do this, we substitute 2 in place of \(x\) wherever \(x\) appears in the expression. This substitution transforms the function into a specific numeric expression: \(g(2) = \frac{2-2}{9+2}\). This process is crucial because it allows us to evaluate the function at a given point, helping to understand how the function behaves for specific inputs.
Simplification
Simplification involves reducing an expression to its simplest form, making it easier to understand and evaluate. After substitution, we have \(g(2) = \frac{2-2}{9+2}\). Simplification starts by calculating each part of the expression separately.
  • The numerator is simplified from \(2-2\) to \(0\).
  • The denominator is simplified from \(9+2\) to \(11\).
Hence, the simplified expression for \(g(2)\) becomes \(\frac{0}{11}\). This step is essential as it breaks down complex expressions into more manageable parts, revealing the expression's true form and helping us to compute the final result efficiently.
Numerator
The numerator is the top part of a fraction. It represents how many parts of the whole are being considered. In our function \(g(2) = \frac{0}{11}\), after substitution and simplification, we identified that the numerator is \(0\).
- Numerators are crucial since they dictate the values of the fraction when comparing it to the denominator. - In this case, because the numerator is zero, it directly affects the overall value of the function, bringing it to zero regardless of what the denominator is. Understanding the numerator's role helps in grasping how fractions work and how changing the numerator affects the outcome of the expression.
Denominator
A denominator is the bottom part of a fraction. It indicates how many equal parts the whole is divided into. For our function when \(x = 2\), the denominator is the part of the function \(9+x\) leading to \(9+2 = 11\).
- The denominator provides the scale for the fraction; it tells "into how many parts" we are dividing. - The value of a fraction is the result of dividing the numerator by the denominator. In our example, even though the denominator is \(11\), the fraction's entire value is determined by the zero numerator, resulting in \(0\), showcasing the intriguing relationship and balance between numerators and denominators in fractions.

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Most popular questions from this chapter

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