Common Types of Mistakes: Incorrectly Assuming Linearity
Why are you confusing me with two meanings of linear?
When you get further along in math (for example, to linear algebra), "linear" has the meaning described at left. It's unfortunate that all functions f(x) = mx+b (for constant coefficients m and b) (which have graphs that are straight lines) are commonly said to be linear functions. We could avoid confusion (now and later), by calling such a function f(x) = mx+b an affine function right from the start. Then it would be perfectly clear to say that f(x) = -2x+3 is an affine function, but not a linear function.
A whole range of common mistakes fall under the category of incorrectly assuming that some function is linear. A function f is linear in the sense described here if both f(a + b) = f(a) + f(b) and f(ca) = cf(a) are always true for any choices of a, b and c. (In the case of functions of one real variable) only functions with graphs that are (subsets of) nonvertical straight lines through the origin are linear in this sense. Any other function can be seen to be nonlinear by giving one counterexample, that is, finding some choices of a and b so that f(a + b) ≠ f(a) + f(b) or some c and a so that f(ca) ≠ cf(a). Unfortunately, a common algebra mistake is to assume that (in particular) f(a + b) = f(a) + f(b), regardless of the function f. It's possible that these mistakes occur because function notation is not understood correctly.
Here are some examples.
Example 1: ("Freshman exponentiation")
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
The mistake is to assume that the function f(x) = x2 is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.
Click to see a counterexample. Click to remove the counterexample.
Click to see a correct expansion. Click to hide the correct simplification.
The same principle applies to other powers of expressions of the form (x + y), for example, 3 or 4. It extends also to negative powers and fractional powers, like square roots.
Example 2: (Reciprocals)
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
The mistake is to assume that the function f(x) = 1/x = x-1 is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.
Click to see a counterexample. Click to remove the counterexample.
There is no correct general simplified expression for the reciprocal of a sum.
Example 3: (Square roots)
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
The mistake is to assume that the function f(x) = √x is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.
Click to see a counterexample. Click to remove the counterexample.
There is no correct general expansion for a square root of a sum.
To summarize the first three examples: Except when n=1, the power function f(x) = xn is not linear.
Logarithm functions are also not linear.
Example 4: (Logarithms)
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
The mistake is to assume that the function f(x) = ln x is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.
Click to see a counterexample. Click to remove the counterexample.
There is no correct general expansion for the logarithm of a sum.
Trigonometric functions are also not linear.
Example 5: (Trigonometric functions)
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
The mistake is to assume that the function f(x) = cos(x) is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.
Click to see a counterexample. Click to remove the counterexample.
The correct expansion is given by the formula for the cosine of a sum:
Click to see the correct sum rule for cosine. Click to hide the correct sum rule for cosine.
Of course, none of the other trigonometric functions are linear either.
These examples are just a small sampling of functions that are sometimes incorrectly assumed to be linear. Don't assume that a function is linear unless it actually is!