91Ó°ÊÓ

Function evaluation refers to the process of determining the output of a function when a specific input is substituted into it. This is a fundamental concept in mathematics, as it allows us to understand how functions behave for different input values. In our exercise, we have two functions: a linear function, \(f(x) = x + 6\), and a quadratic function, \(g(x) = x^2 - 5\). To find values like \(f(g(0))\), \(g(f(0))\), \(g(g(2))\), and \(f(g(7))\), we must first evaluate the inner function with a given input and then use that result as the input for the outer function.

For example, to find \(f(g(0))\):

• First, substitute 0 into \(g(x)\) to find \(g(0) = 0^2 - 5 = -5\).
• Then, use \(-5\) as the input in \(f(x)\): \(f(-5) = -5 + 6 = 1\).

This process ensures we correctly evaluate functions step by step and understand the progression from one calculation to the next.

A quadratic function is one of the most common types of polynomial functions, characterized by the square of the input variable. It takes the general form \(g(x) = ax^2 + bx + c\). In our exercise, the quadratic function is \(g(x) = x^2 - 5\), which simplifies the general form to where \(a = 1\), \(b = 0\), and \(c = -5\).

Key properties of quadratic functions:

• They graph as parabolas, which can open upwards or downwards depending on the sign of \(a\).
• The vertex of the parabola represents the maximum or minimum point.
• They have a symmetric axis that passes through the vertex.

For the quadratic function \(g(x)\) in our exercise, the process of calculation involves squaring the input and subtracting 5 from the squared value. When evaluating \(g(g(2))\), we substitute into itself to show this compound nature of function evaluation. Understanding how terms like \(x^2\) influence the function's curve is crucial for analyzing quadratic behaviors.

Linear functions are the simplest type of polynomial functions. They form the equation \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. In this exercise, the linear function is given by \(f(x) = x + 6\). Here, the slope \(m\) is 1, and the y-intercept \(b\) is 6.

Characteristics of linear functions include:

• Graphing as straight lines.
• Consistent slope, meaning no curves or sharp angles.
• Directly proportional relationship between \(x\) and \(f(x)\).

In the context of the exercise, evaluating \(f(x)\) involves simply adding 6 to the input value. When composing with another function like \(g\), the order of operations becomes particularly important. For example, in \(f(g(7))\), we first find the result of \(g(7)\) to get 44, then add 6 through the linear function to achieve a final result of 50. This sequence illustrates how linear transformations straightforwardly adjust values within their applied range.