Write a linear function $f$ with the given values.
$f(x) = \square$
A linear function is a mathematical function that graphs as a straight line, typically written in the form \(f(x) = mx + b\), where \(m\) represents the slope (the rate of change) and \(b\) represents the y-intercept (the value of the function when \(x=0\)). To uniquely determine a specific linear function, one generally requires two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) that lie on the line. Since the problem as stated does not provide these values, we will demonstrate the process of determining such a function using a general example where we are given two points: \((x_1, y_1)\) and \((x_2, y_2)\).
To find the linear function passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we first calculate the slope \(m\). The slope is defined as the ratio of the change in the y-values to the change in the x-values:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Once the slope \(m\) is determined, we can find the y-intercept \(b\) by substituting one of the points into the equation \(y = mx + b\) and solving for \(b\):
\[b = y_1 - m \cdot x_1\]
With both \(m\) and \(b\) found, the function is uniquely defined as \(f(x) = mx + b\).
Suppose we are given two points: \((1, 3)\) and \((2, 5)\).
First, we calculate the slope:
\[m = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2\]
Next, we solve for the intercept \(b\) using the point \((1, 3)\):
\[3 = 2(1) + b\]
\[3 = 2 + b \implies b = 1\]
Thus, the linear function is \(f(x) = 2x + 1\). In the absence of specific data in your request, this illustrates the methodology required to solve for any linear function when provided with sufficient constraints.
In real-world applications, linear functions are used to model relationships with a constant rate of change. For example, if a taxi service charges a base fee of \$5 and then \$2 for every mile traveled, the total cost \(f(x)\) as a function of the number of miles \(x\) can be modeled by the linear function \(f(x) = 2x + 5\). By identifying the fixed costs (the y-intercept) and the variable costs per unit (the slope), we can accurately predict costs or outcomes for various scenarios.