This posts contains a Teaching Explanation.

You can buy *Calculus *by Stewart here.

**Why You Should Trust Me:** I’m Dr. Fred Zhang, and I have a bachelor’s degree in math from Harvard. I’ve racked up hundreds and hundreds of hours of experience working with students from 5^{th} grade through graduate school, and I’m passionate about teaching. I’ve read the whole chapter of the text beforehand and spent a good amount of time thinking about what the best explanation is and what sort of solutions I would have wanted to see in the problem sets I assigned myself when I taught.

**Question:** If $f(z) = z - √(2-z)$ and $g(u) = u - √(2-u)$ is it true that f =g?

**Page in 8th Edition:** 19

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

**Homework Answer:** Because the equation for f(x) and g(u) are the same, this means that for all valid inputs for function f, the function f and g give the same output.

In other words, for all valid z, $f(z) = z - √(2-z) = g(z)$.

**Motivated Answer:**

This question is asking if f = g. What does it mean for two functions to be equal? We know that 2 = 2, and if someone asks, does 2=3? We know the answer is “no”, but does f = g?

Remember, functions take in inputs, and spit out outputs. Two functions f and g are only equal if they always give you the same output no matter what the input is.

Let’s see what happens if we put in any valid input z into f. We get $f(z) = z - √(2-z)$. Now let’s put that same z into g, and we get $g(z) = z - √(2-z)$. These two are the same, and so f and g are the same.

This question is a bit of a trick. The textbook writes $g(u) = u - √(2-u)$, but they could have just written $g(x) = x - √(2-x)$. This would have made it much more clear that f = g.

There are two key learning points to take away:

- Two functions can be the same even if the equations look different written out.
- The above point is NOT true in reverse: If you substitute the same variable z into two functions’ equations, and can get the equations to look the same, then the functions are the same.

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