Stewarts Calculus 8th Edition Section 1.1 Question 1

This post contains a Teaching Explanation. You can buy Calculus by Stewart here.

Why You Should Trust Me:

My name is Dr. Fred Zhang, and I have a bachelor’s degree in math from Harvard. I have racked up hundreds of hours of involvement working with students from grade 5 through graduate school, and I love teaching. I have read the whole chapter of the text beforehand and spent a reasonable amount of time thinking about what the best explanation is and what sort of answers I would have wanted to see in the problem sets I assigned myself while I was teaching.

Problem:

If \$f(z) = z -√(2-z)\$ and \$g(u) = u -√(2-u)\$ is it a fact that f =g?

Section in the Eighth Volume: 19

Quick Answer: Yes, it is true that f=g because the equation for g is precisely the same as that for f, except with x replaced by u.

Homework Answer: Since the equation for f(x) and g(u) is invariable, it means that for all valid inputs for function f, the function f and g give the same result.

That is to say, for all valid z, \$f(z) = z – √(2-z) = g(z)\$.

The question is asking if f = g.

What exactly does it mean for two functions to be the same?

We are aware that 2 = 2, and if anyone asks, does 2=3?

We know the result is “no,” but does f = g?

Don’t forget, functions take in inputs and take out outputs. Two functions, f, and g are only the same if they always give you the same result no matter what the input is.

Let’s see what happens if we put any valid input z into f.

We obtain \$f(z) = z – √(2-z)\$.

Let us put that same z into g, and we get \$g(z) = z – √(2-z)\$.

These two are the same, so f and g are the same. This question is a bit of a trial. In the textbook: \$g(u) = u – √(2-u)\$, but they could have written \$g(x) = x – √(2-x)\$.

This would have made it more clear that f = g.

Two key learning points to take note of:

• As much as the equations look differently written out, two functions can be the same.
• The above point is NOT valid in reverse: If you substitute the same variable z into two functions’ equations and get the equations to look alike, then the functions are alike.

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