Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(0)\)

Step-by-Step Solution

Answer

The value of the function \((f-g)(0)\) is 3

Step 1: Evaluate Function f at x=0

To evaluate function f at x=0, replace x with 0 in the expression for f(x): \[f(0)=2(0)+1=1\]

Step 2: Evaluate Function g at x=0

To evaluate function g at x=0, replace x with 0 in the expression for g(x): \[g(0)=(0)^{2}-2=-2\]

Step 3: Compute The Difference Between The Two Functions

The difference between the two functions is given by \[f(0)-g(0)=1-(-2)=3\]

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, where we determine the output of a function for a given input. When we have a function like \( f(x) = 2x + 1 \) and we want to find its value at a specific point, say \( x = 0 \), we substitute that value in place of \( x \).
- For example, evaluating \( f(0) \) entails replacing \( x \) with 0: - Thus, \( f(0) = 2(0) + 1 \), which simplifies to 1.
Similarly, for \( g(x) = x^2 - 2 \), evaluating at \( x = 0 \) involves the substitution \( g(0) = (0)^2 - 2 \), resulting in -2.
Whenever evaluating functions, it's important to carefully follow these substitution steps, ensuring accuracy in operations and handling negative signs when needed.

Difference of Functions

The difference of functions involves subtracting one function from another. It is a useful operation when comparing outputs of two separate functions.
To find the difference, say \((f-g)(x)\), you simply take the output of \( f(x) \) and subtract \( g(x) \) at the same \( x \) value.
- Using our functions \( f(x) = 2x + 1 \) and \( g(x) = x^2 - 2 \), to calculate \( (f-g)(0)\), we first find: - \( f(0) = 1 \) - \( g(0) = -2 \)
Then, subtract the results: - \( f(0) - g(0) = 1 - (-2) \), which results in 3.
Subtraction becomes addition with a negative sign, so care must be taken to manage signs properly for correct results.

Polynomial Functions

Polynomial functions are a broad class of functions that involve terms of variables raised to whole number powers and coefficients. Simple polynomials can be linear, such as \( f(x) = 2x + 1 \), which has a power of one. Quadratic functions, like \( g(x) = x^2 - 2 \), have a variable raised to the power of two.
- Key characteristics of polynomial functions include: - Their graphs are smooth and continuous lines or curves. - The degree of the polynomial determines its shape and complexity.
For example, a linear polynomial like \( f(x) \) is a straight line, while a quadratic like \( g(x) \) forms a parabola.Understanding polynomial functions helps in evaluating and comparing their values, and operations such as finding differences (e.g. \((f-g)(x)\)) become straightforward once we recognize their forms and behaviors.