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Chapter 7: Problem 9

For each function, evaluate the given expression. \(f(x, y)=\sqrt{99-x^{2}-y^{2}}\), find \(f(3,-9)\)

Short Answer

Expert verified
The value of \( f(3, -9) \) is 3.

Step by step solution

01

Function Substitution Setup

The given function is \( f(x,y) = \sqrt{99-x^2-y^2} \). We need to find \( f(3, -9) \). This involves substituting \( x = 3 \) and \( y = -9 \) into the function.
02

Substitute the Values

Substitute \( x = 3 \) and \( y = -9 \) into the expression inside the square root: \( 99 - 3^2 - (-9)^2 \).
03

Calculate Squares

Calculate \( 3^2 \) which is 9 and \( (-9)^2 \) which is 81. So, the expression becomes \( 99 - 9 - 81 \).
04

Simplify the Expression

Simplify the inside of the square root: \( 99 - 9 - 81 = 9 \).
05

Compute the Square Root

Take the square root of 9, which is 3.

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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 a fundamental concept in calculus and algebra.It involves finding the output of a function given specific input values.In this exercise, we evaluated the function \( f(x, y) = \sqrt{99 - x^2 - y^2} \) by substituting \( x = 3 \) and \( y = -9 \).To effectively evaluate functions:
  • Identify the given function and its variables.
  • Substitute the provided values into the function.
  • Perform any necessary calculations to simplify the expression.
  • Compute to find the final result.
This practice helps build a strong foundation for solving more complex problems.It enhances analytical skills and reinforces understanding by applying theoretical knowledge to practical examples.
By mastering function evaluation, students can confidently tackle various mathematical challenges.
Step-by-Step Mathematical Solution
Solving mathematical problems by following step-by-step solutions can greatly simplify complex tasks.Breaking down the calculation of \( f(3, -9) \) into manageable steps makes it easier to follow.Let's revisit each stage of the solution for clarity.
  • First, we substitute \( x = 3 \) and \( y = -9 \) into the function \( \sqrt{99 - x^2 - y^2} \).
  • Next, we replace \( x\) and \( y\) in the expression: \( 99 - 3^2 - (-9)^2 \).
  • We then calculate \( 3^2 \) which is 9, and \((-9)^2\) which is 81.
  • The expression simplifies to \( 99 - 9 - 81 \).
  • We simplify further to find it equals 9.
Solving in this manner helps prevent errors and ensures each calculation step is correct.It is especially beneficial for understanding and mastering similar problems in homework and exams.
Square Root Calculation
Calculating the square root is a key part of many mathematical solutions, including this exercise.It involves finding a number which, when multiplied by itself, gives the original number.In this problem, after simplifying the expression to 9, we compute the square root.To understand square root calculations:
  • Acknowledge that the square root of a number \( a \), denoted as \( \sqrt{a} \), is a value \( b \) such that \( b^2 = a \).
  • For positive numbers like 9, the square root is straightforward: \( \sqrt{9} = 3 \), because \( 3 \times 3 = 9 \).
  • Be cautious with perfect squares and familiar numbers for quick recall.
Understanding these principles is crucial for solving both basic arithmetic problems and more advanced calculus questions.

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