Some algebra mistakes appear to be based on misunderstanding function notation.

A function is a rule that assigns to each (input) element of one set (the domain) an (output) element of another set (the range) in such a way that the output value is always the same for a given fixed input value (graphically, a function passes the vertical line test). (There are more technical ways to define function, but this rule definition works for our purposes here.) The way in which the rule for a given function is represented varies from function to function, and this may be part of the problem in recognizing that all functions share the properties stated in the definition.

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.

Example 1: Determine f(x+h) for

Mistake:

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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.

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Check your understanding by determining f(x-h) for

and then check your answer:

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There are many ways that different functions are written. Here are some examples:

f(x) = √x

f(x) = ln(x)

f(x) = e^x

f(x) = sin(x)

f(x) = sin^-1(x)

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^x); and sometimes symbols are compounded to represent functions (like the symbol sin^-1 to represent the inverse sine function). But however a function is written, it is still a function, that is, a rule that takes inputs and produces outputs. That means you can't treat the symbols that represent functions as variables that can be manipulated like other parts of algebra. Algebra done with a function must follow the rules that apply to that function.

Here (I hope) is an obvious mistake.

Example 2: For the function f(x) = x+1,

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Cancellation of common factors is allowed. But the function notation f(2) does not mean "f multiplied by 2". There is no common factor. The function notation f(2) does mean "apply the function (that is, rule) f to the input 2". The "cancellation" shown above is not allowed.

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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.

Example 3:

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The "ln"s can not be cancelled, since they are not factors. The notation indicates that the natural logarithm function is to be applied to the quantities x+1 and x², respectively.

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 common factor that can be cancelled.

What you want to do, instead, is to undo the effect of the natural logarithm function. If you want to undo a square root, you square. That's because (√x)² = x, that is, the squaring function inverts the square root function. To undo the natural logarithm, you use it's inverse function, which is e^x. This means that e^ln(A) = A for any allowable quantity A (namely, A>0).

So, starting with ln(x+2) = ln(5), apply the exponential function to both sides to get

e^ln(x+2) = e^ln(5), that is, x+2 = 5 (using the property e^ln(A) = A).

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.

Example 4:

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The notation f^-1(x) does not indicate the reciprocal of the function f(x). Instead it indicates the inverse function of f(x). As mentioned earlier, that's the function that undoes f(x). The inverse of the square root function is the squaring function, not the reciprocal of the square root function.

For many functions f(x), the inverse function f^-1(x) has its own distinct notation. For example, the natural logarithm and exponential functions are inverses of each other. Some functions, like the inverse trigonometric functions, are sometimes represented using the general inverse notation, as above.

Example 5:

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The notation cos^-1(x) means the inverse cosine of x. Recall that cos^-1(x) = θ means that cos(θ) = x and 0 ≤ θ ≤ π. So the task here is to find an appropriate angle whose cosine is equal to 0.

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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.