91Ó°ÊÓ

Function evaluation is a crucial concept in mathematics, especially when dealing with functions and their graphs. It involves substituting a specific value of the variable into the function and simplifying to find the output. In our example, we were asked to check if the point \( (1,1) \) is on the graph of \( g(x) = \frac{3x - 1}{x^2 + 1} \). To do this, we substitute \( x = 1 \) into the function: \[ g(1) = \frac{3(1) - 1}{1^2 + 1} = \frac{2}{2} = 1 \]. The output is 1, which matches the y-coordinate of the point \( (1, 1) \). This means the point \( (1,1) \) is indeed on the graph of the function.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In our case, we are working with the Cartesian coordinate system, where we have two axes (x and y) that intersect at the origin (0,0). Points in this system are represented as \( (x, y) \) pairs. To determine if a point lies on the graph of a function, we check if substituting the x-coordinate into the function results in the y-coordinate. We verified this with \( (1,1) \) and the function \( g(x) \) by showing that \[ g(1) = 1 \]. This aligns the concept of evaluating functions with coordinate geometry, providing a visual confirmation on the graph.

Function graphing involves plotting the values of a function on a coordinate plane. Each point on the graph represents an \( (x, y) \) pair such that \( y = f(x) \). For our function \( g(x) = \frac{3x - 1}{x^2 + 1} \), we can plot several points by evaluating the function at different x-values. Graphing helps in visualizing the behavior of functions and understanding their properties. For example, by plotting several points, we can see patterns such as where the function increases or decreases, its intercepts, and overall shape. Specifically, we confirmed that the point \( (1,1) \) lies on the graph by showing \[ g(1) = 1 \]. This one point tells us that when x is 1, the function yields a y-value of 1, thus placing the point on the graph.

Algebraic manipulation is the process of rearranging, simplifying, or transforming mathematical expressions. In the context of this exercise, we performed algebraic manipulation by substituting \( x = 1 \) into the function \( g(x) \) and simplifying: \[ g(1) = \frac{3(1) - 1}{1^2 + 1} = \frac{2}{2} = 1 \]. This step is all about applying basic algebraic rules to evaluate functions, solve equations, and simplify expressions. Through algebraic manipulation, we can confirm that our function's output matches the y-coordinate of the given point, proving its presence on the graph.