Problem 7 - Calculus: One and Several Variables

Question

Set \(f(x)=2 x^{2}-3 x+1\) and \(g(x)=x^{2}+1 / x\) Calculate the indicated value. $$(f \circ g)(1)$$

Step-by-Step Solution

Answer

The value of the composition function \(f(g(1))\) is 3.

Step 1: Calculate the value of g(1)

Substitute \(x=1\) in the function \(g(x)=x^{2}+1 / x\) to get \(g(1) = 1^{2} + \frac{1}{1} = 2\)

Step 2: Substitute g(1) into f(x)

Substitute \(g(1)=2\) in the function \(f(x)=2 x^{2}-3 x+1\)

Step 3: Evaluate f(g(1))

Substitute \(2\) in the function to get \(f(g(1))=2 * 2^{2} -3*2 + 1 = 8 - 6 + 1 = 3\)

Key Concepts

Composite Functions

Understanding composite functions is crucial for mastering more complex mathematical relationships. A composite function, denoted as \(f \circ g)(x)\), results from the combination of two functions, where the output of the second function \(g(x)\) becomes the input for the first function \(f(x)\). This process is akin to a relay race, where the baton \(g(x)\) is passed from one runner \(g\) to another \(f\).

To evaluate \(f \circ g)(x)\), you start by finding the output of \(g(x)\), which you then use as the input for \(f(x)\). In our example, we computed \(g(1)\) first, obtaining a value of 2, which served as the input to find \(f(2)\). This sequential process is what makes composite functions valuable, allowing for the construction of new functions that can express more complicated scenarios.

Polynomial Functions

Polynomial functions are like numerical symphonies; they are composed of terms that include variables raised to whole number exponents and their coefficients. For example, \(f(x) = 2x^2 - 3x + 1\) is a polynomial of degree two because the highest power of \(x\) is two, referred to as a quadratic polynomial. Here are the basics to recognize:

• The coefficient of \(x^2\) is 2.
• The coefficient of \(x\) is -3.
• The constant term is 1.

Such functions are fundamental in mathematics due to their nice properties, such as continuity and differentiability throughout their entire domain. When evaluating polynomials, the order of operations (PEMDAS) serves as a guide—powers and multiplications come before additions and subtractions.

Function Evaluation

Function evaluation might seem daunting at first, but it's simply the process of determining the output of a function given a specific input. If you regard a function as a vending machine, inputting \(x\) is like inserting your money and getting a specific output, \(f(x)\), which is your chosen snack.

To evaluate a function like \(f(x) = 2x^2 - 3x + 1\), you replace the variable \(x\) with a particular value. If \(x=2\), you would calculate \(2(2)^2 - 3(2) + 1\). Following the order of operations, square first, then multiply, followed by subtraction and addition. This process simplifies to \(2\times4 - 3\times2 + 1\), which further simplifies to 8 - 6 + 1, equaling 3. Function evaluation is the heart of working with functions, and it's the tool we use to understand how functions behave for different inputs.