Tuesday 11 February class


• Expressing one variable as a function of another

Examples 2.20, 2.21, 2.22

• The square root principle (used in Example 2.22 in class):

When solving an equation of the form (something), we use the Square Root Principle:

If , then

$latex x = \pm\sqrt{c}

Note: there is no “missing step”. We do not “take the square root of both sides”. What is wrong with doing that has to do with the fact that \sqrt{x^2} is NOT necessarily x. (And yes, I know that some teachers tell you to “square root both sides”, ugh, but it is still wrong, and will cause you trouble later on.)

Here is a video that explains it correctly. (You’ll have to sit through an annoying ad first, but the video is good and is totally mathematically correct.)

If you write the incorrect version, it is an error (that’s the main consideration!) and will cause you to have fewer points on that problem (if that’s all you care about). So correct this in your thinking if necessary!


• Functions given by formulas

Example 3.1: computing various values of the function, and thinking about the domain of the function (what values cause difficulties? Where is the function defined?)

We went ahead and found the domains for each of the functions in Ex. 3.1

(a) A polynomial function 3x+4:The domain is R

(b) A radical function \sqrt{x^2-3} : we need the radicand to be nonnegative x^2-3\ge 0 ,  so the domain is found by solving an inequality. We solved it the same way we solved absolutevalue inequalities before: the domain is (-\infty, -\sqrt{3}]\cup[\sqrt{3}, \infty)

(c) A piecewise-defined function: we find the domain by fitting together the various intervals where the function is defined

(d) A rational function: we need the denominator not to be 0, so the domain is all real numbers except -3


Example 3.5: finding the domains of more functions of these four types.


• Computing values of the function for more complicated inputs, and computing a difference quotient.

Example 3.2(e-k)

See also Example 3.3(a)

Homework: see the previous post.


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