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Chapter 6: Problem 4

For \(f(x, y)=\log _{10}(x+y)+3 x^{2},\) find \(f(3,7), f(1,99)\) \(\operatorname{and} f(2,-1)\)

Short Answer

Expert verified
f(3, 7) = 28, f(1, 99) = 5, f(2, -1) = 12.

Step by step solution

01

Identify the function

Given function is \( f(x, y) = \ \ \log_{10}(x + y) + 3x^2 \). This will be used to find the values of \(f(3, 7)\), \(f(1, 99)\), and \(f(2, -1)\).
02

Calculate \(f(3, 7)\)

Substitute \(x = 3\) and \(y = 7\) into the function: \( f(3, 7) = \ \ \log_{10}(3 + 7) + 3(3^2) \). Simplify the expression: \(f(3, 7) = \ \ \log_{10}(10) + 3 \times 9 = 1 + 27 = 28\).
03

Calculate \(f(1, 99)\)

Substitute \(x = 1\) and \(y = 99\) into the function: \( f(1, 99) = \ \ \log_{10}(1 + 99) + 3(1^2) \). Simplify the expression: \(f(1, 99) = \ \ \log_{10}(100) + 3 \times 1 = 2 + 3 = 5\).
04

Calculate \(f(2, -1)\)

Substitute \(x = 2\) and \(y = -1\) into the function: \( f(2, -1) = \ \ \log_{10}(2 + (-1)) + 3(2^2) \). Simplify the expression: \(f(2, -1) = \ \ \log_{10}(1) + 3 \times 4 = 0 + 12 = 12\).

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Key Concepts

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

logarithmic functions
Logarithmic functions are the inverses of exponential functions. The logarithm of a number is the exponent to which a base, usually 10 or e, must be raised to produce that number. For example, \(\log_{10}(100) = 2\) because \(10^2 = 100\). The base-10 logarithm, also known as the common logarithm, is often written simply as \(\log(x)\).
  • In the given problem, we work with the base-10 logarithm, \(\log_{10}(x+y)\).
  • Understanding how to use and simplify logarithms is essential for evaluating functions involving logarithms.
Next, we'll delve into how to evaluate our function using specific values for x and y.
function evaluation
Evaluating a function means finding the value of the function for specific inputs. Here, we are given a function \(f(x, y) = \log_{10}(x + y) + 3x^2\) and asked to calculate its values for different pairs of x and y.
  • To evaluate \(f(3,7)\), substitute \(x = 3\) and \(y = 7\) into the function.
  • Similarly, for \(f(1,99)\), substitute \(x = 1\) and \(y = 99\).
  • Lastly, for \(f(2,-1)\), substitute \(x = 2\) and \(y = -1\).
Substituting these values into the given function, we can solve step-by-step as detailed below.
step-by-step solution
Let's walk through the calculation of each value step by step to ensure we understand the process.
First, identify the given function: \(f(x, y) = \log_{10}(x + y) + 3x^2\).
  • Step 2: Calculate \(\mathbf{f(3, 7)}\):\

    Substitute \(x = 3\) and \(y = 7\) into the function: \(f(3, 7) = \log_{10}(3 + 7) + 3(3^2)\).
    Simplify: \(f(3, 7) = \log_{10}(10) + 3(9) = 1 + 27 = 28\).
  • Step 3: Calculate \(\mathbf{f(1, 99)}\):\

    Substitute \(x = 1\) and \(y = 99\): \(f(1, 99) = \log_{10}(1 + 99) + 3(1^2)\).
    Simplify: \(f(1, 99) = \log_{10}(100) + 3(1) = 2 + 3 = 5\).
  • Step 4: Calculate \(\mathbf{f(2, -1)}\):\

    Substitute \(x = 2\) and \(y = -1\): \(f(2, -1) = \log_{10}(2 + -1) + 3(2^2)\).
    Simplify: \(f(2, -1) = \log_{10}(1) + 3(4) = 0 + 12 = 12\).
By following these steps, we can confidently evaluate the function for different sets of inputs.

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Most popular questions from this chapter

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