![]() The affine function $f(x)=ax+b$ is illustrated by its graph, which is the green line. The rebellious view of the linear function is to call any function of the formĪ linear function, since its graph is a line.Īn affine function of one variable. The rebellious view of the linear function ![]() The other requirement for a linear function is that applying $f$ to the sum of two inputs $x$ and $y$ is the same thing as adding the results from being applied to the inputs individually, i.e., $f(x+y)=f(x)+f(y)$. A linear function must satisfy $f(cx)=cf(x)$ for any number $c$. Therefore, the doubling requirement means $f(0)=2f(0)$, so $f(0)$ is a number that is the same if you double it i.e., $f(0)=0$.īy the way, for a linear function, this property must be satisfied for any number, not just the number 2. This follows from the fact that if you double zero, you get zero back. To satisfy this doubling requirement, we must have $f(0)=0$. If $f(x)=ax$, then $f(2x)=2ax$ and $2f(x)=2ax$, so this requirement is satified. We can write this requirement for a linear function $f$ asįor any input $x$. It's easy to see that the function $g(x)$ fails this test. One important requirement for a linear function is: doubling the input $x$ must double the function output $f(x)$. Instead, we require certain properties of the function $f(x)$ for it to be linear. Why this insistence that $f(0)=0$ for any linear function $f$? The reason is that in mathematics (other than in elementary mathematics), we don't define linear by the requirement that the graph is a line. Is not a linear function, as $g(0) \ne 0$. By this strict definition of a linear function, the function This fact is the reason the graph of $f$ always goes through the origin. One important consequence of this definition of a linear function is that $f(0)=0$, no matter what value you choose for the parameter $a$. The parameter $a$ is the slope of the line, as illustrated by the shaded triangle. You can change $f$ by typing in a new value for $a$, or by dragging the blue point with your mouse. Since $f(0)=a \times 0 =0$, the graph always goes through the origin $(0,0)$. The linear function $f(x)=ax$ is illustrated by its graph, which is the green line. The graph of $f$ is a line through the origin and the parameter $a$ is the slope of this line.Ī linear function of one variable. Where the parameter $a$ is any real number. In one variable, the linear function is exceedingly simple. Then, we discuss the rebellious definition of a linear function, which is the definition one typically learning in elementary mathematics but is a rebellious definition since such a function isn't linear. ![]() We first outline the strict definition of a linear function, which is the favorite version in higher mathematics. Fortunately, the distinction is pretty simple. Unfortunately, the term “linear function” means slightly different things to different. Because it is so nice, we often simplify more complicated functions into linear functions in order to understand aspects of the complicated functions. It's one of the easiest functions to understand, and it often shows up when you least expect it. The linear function is arguably the most important function in mathematics.
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