Algebra Facts
Common Types of Mistakes: Misunderstanding Function Notation |

Some algebra mistakes appear to be based on misunderstanding function notation.
A When a function is expressed by a formula f(x) it is important to understand that the variable "x" is simply a placeholder for whatever input value(s) the function will be evaluated at, including possibly expressions involving x itself.
Mistake:
Click to see the mistake in red. Click to remove the red highlighting. To find f(x+h) you must replace x wherever it is found in the formula for f by exactly the expression x+h. That is what the function notation tells you to do. Click to see a correct expression. Click to hide the correct expression.
Check your understanding by determining f(x-h) for
and then check your answer:
Click to check your answer. Click to remove the answer. There are many ways that different functions are written. Here are some examples: f(x) = √x f(x) = ln(x)
f(x) = e f(x) = sin(x)
f(x) = sin
Sometimes there is a special symbol (like √); sometimes a special sequence of letters that stands for a longer full name (like ln or sin); sometimes the variable x appears as an exponent (as in e Here (I hope) is an obvious mistake.
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Cancellation of Click to see a correct evaluation. Click to hide the correct evaluation.
There is no cancellation, just function evaluation. (Remember that f(x) = x+1 in this example.) Exactly the same type of mistake is made in the following example.
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The "ln"s can Incorrect Thinking About Function Notation When Solving Equations Can Lead to Mistakes Elsewhere Suppose you want to solve the equation ln(x+2) = ln(5). You next write down x+2 = 5 (and solve to get x=3, the correct answer). How did you get from ln(x+2) = ln(5) to x+2 = 5? Did you "cancel" the natural log from both sides? That's not the right way to think about it, because the ln is not a What you want to do, instead, is to So, starting with ln(x+2) = ln(5), apply the exponential function to both sides to get e Notice that the natural logs were not cancelled, they were undone. The result is the same, so what's the problem? Well, the problem is that thinking of a piece of function notation as something that can be "cancelled" (as a common factor) leads to mistakes like the one in Example 3 above.
The convention for indicating a general inverse function is sometimes misunderstood.
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The notation f
For many functions f(x), the inverse function f
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The notation cos Click to see a correct evaluation. Click to hide the correct evaluation.
These are just a few examples of mistakes caused by not properly understanding function notation. It's important to work on understanding function notation so you can avoid mistakes like these. |

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