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Chapter 29: Problem 1

Evaluate the given functions. $$f(x, y)=3 x^{2} y-y^{3}, \text { find } f(-1,4) \text { and } f(2,-3)$$

Short Answer

Expert verified
\(f(-1,4) = -52\) and \(f(2,-3) = -9\).

Step by step solution

01

Plug in Values for First Set

Substitute the values \(x = -1\) and \(y = 4\) into the function \(f(x, y) = 3x^2y - y^3\).
02

Evaluate the Expression for First Set

Calculate \(3(-1)^2(4) - 4^3 = 3(1)(4) - 64 = 12 - 64 = -52\).
03

Plug in Values for Second Set

Substitute the values \(x = 2\) and \(y = -3\) into the function \(f(x, y) = 3x^2y - y^3\).
04

Evaluate the Expression for Second Set

Calculate \(3(2)^2(-3) - (-3)^3 = 3(4)(-3) + 27 = -36 + 27 = -9\).

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Key Concepts

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

Function Evaluation
Function evaluation is the process of determining the output of a multivariable function given specific input values. In this exercise, we are given a function of two variables, \( f(x, y) = 3x^2y - y^3 \), and we need to find its value for certain inputs.
To evaluate a function:
  • Identify the variables in the function, here being \(x\) and \(y\).
  • Replace these variables with the given numbers.
  • Use algebraic operations to simplify and find the result.
This direct substitution technique allows us to understand how changes in input affect the function's output. It's a fundamental skill in calculus and algebra, helping us analyze and predict the behavior of functions.
Step-by-Step Solution
Step-by-step solutions make complex problems easier by breaking them down into manageable parts.
For instance, in evaluating the function \( f(x, y) = 3x^2y - y^3 \) at specific points, we evaluate each input separately.
  • Step 1 substitutes \( x = -1 \) and \( y = 4 \) into the function, replacing \(x\) and \(y\) in the expression.
  • Step 2 calculates the expression: using simple arithmetic, determining that \( f(-1, 4) = -52 \).
  • Step 3 substitutes \( x = 2 \) and \( y = -3 \).
  • Step 4 simplifies this expression to find \( f(2, -3) = -9 \).
This approach ensures clarity by making each change and operation explicit. Following these steps systematically allows for an organized and efficient evaluation process.
Algebraic Substitution
Algebraic substitution is a pivotal technique used in function evaluation and problem-solving. It involves replacing variables in a mathematical expression with specific values.
In our problem, the substitution of \( x = -1 \), \( y = 4 \) and \( x = 2 \), \( y = -3 \) into the function \( f(x, y) = 3x^2y - y^3 \) helps determine the function's output for these exact inputs.
This method:
  • Transforms abstract expressions into numerical calculations.
  • Facilitates checking of solutions by easily substituting different values.
  • Helps understand the effect of variables on the outcome.
Overall, algebraic substitution is a straightforward yet powerful tool that simplifies evaluating functions and solving equations.
Mathematical Expressions
Mathematical expressions combine numbers, variables, and operators to describe a specific quantity or relationship.
In the given function \( f(x, y) = 3x^2y - y^3 \), each component plays a role:
  • The term \( 3x^2y \) indicates a multiplication of squares and linear terms, transforming with different \(x, y\) values.
  • \(-y^3\) represents a cubic transformation, typically growing larger in magnitude faster than linear terms.
Expressions require careful manipulation and arithmetic to evaluate correctly, especially as variables change. Understanding how to simplify expressions lays the foundation for more advanced algebra and calculus principles.

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