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Function evaluation is an essential mathematical skill that involves finding the output of a function for a particular input value. In this context, you can think of a function as a machine that processes inputs to produce outputs.
To evaluate a function, you simply replace the function's variable with the specific input value you are given.
This substitution allows you to calculate the function's output. In our exercise, you start by evaluating the linear function \(g(x) = 3x - 5\). You substitute \(x = 1\) into the function to find \(g(1)\). This computation gives you \(g(1) = 3(1) - 5 = -2\). Next, you need to evaluate the quadratic function \(f(x) = x^2 - x + 4\). Instead of a direct input, you use the result of the first evaluation \(-2\) (which is \(g(1)\)) as the new input for \(f\), calculating \(f(-2)\). This method of plugging one function's output into another is called composition of functions. The process results in \(f(-2) = (-2)^2 - (-2) + 4 = 8\).

Quadratic functions are a class of functions that can be represented by the equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The graph of a quadratic function is a parabola, which may open upwards or downwards depending on the sign of \(a\).
The vertex of the parabola provides a point of maximum or minimum value of the function. The function \(f(x) = x^2 - x + 4\) in our exercise is an example of a quadratic function.Breaking it down:- The term \(x^2\) is the quadratic term, responsible for the curved shape of the graph.- The term \(-x\) is the linear term, contributing to the tilt of the parabola.- The constant term \(+4\) moves the graph vertically, shifting the entire parabola up by 4 units.When evaluating a quadratic function like \(f(g(1)) = f(-2)\), calculations involve applying the input to each term separately, then summing the results. This type of function is useful in modeling situations where the relationship between variables squares or multiplies results disproportional to each increase of input.

A linear function describes a straight-line relationship between two variables and is expressed generally as \(g(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It defines how much \(y\) changes with one unit increase in \(x\). Linear functions are easy to identify due to their simple form and linear graph.In the given exercise, \(g(x) = 3x - 5\) is a linear function:- The coefficient \(m = 3\) suggests that for every increase of 1 in \(x\), \(g(x)\) increases by 3 units.- The constant \(-5\) serves as the y-intercept, showing where the line crosses the y-axis when \(x=0\).Linear functions are foundational for understanding other types of functions as they represent the simplest form of change with a constant rate. In practical terms, these functions model countless situations, from calculating costs or speeds to forecasting trends. Evaluating these functions is straightforward, as it involves direct substitution of the input value into the equation, followed by basic arithmetic operations.