Precalculus I Class 04/24
Evaluate y = e^(-x^2) for x = 1, 2, 3.
Note that -x^2 means -(x^2) by the order of operations. It does not mean (-x)^2.
| x | u = -x^2 | e^u = e^(-x^2) |
| 1 | -1 | .3678 |
| 2 | -4 | .0183 |
| 3 | -9 | .000123 |
The process we followed is to first evaluate the 'intermediate' function u = -x^2, then express the original function as e^u, plugging in the resulting values of u.
So e^(-x^2) = e^u, with u = -x^2.
Show that we can also write this function as f(g(x)), with g(x) = -x^2 and f(z) = e^z.
If f(z) = e^z then f(g(x)) = e^g(x).
Substituting -x^2 for g(x) we get
f(g(x)) = e^(-x^2).
Evaluate y = (e^(-x))^2 for x = 1, 2, 3.
| x | u = e^-x | y = u^2 = (e^(-x))^2 |
| 1 | .3678 | .1353 |
| 2 | .1353 | .0183 |
| 3 | .0497 | .00247 |
Following the order of operations we evaluate the expression inside parentheses first. This expression is e^(-x).
We see that the evaluation of the function can be broken into steps, first evaluating u = e^-x then squaring the result.
How do we write this function y = (e^(-x))^2 as a composite in the form f(g(x))?
The first thing that happens to x in (e^(-x))^2 is e^(-x).
The first thing that happens to x in f(g(x)) is g(x).
So g(x) = e^(-x).
The next thing that happens in (e^(-x))^2 is that the result is squared.
The next thing that happens in f(g(x)) is that f is applied to the result.
So f is the squaring function. Using 'dummy' variable z we write this as f(z) = z^2.
Evaluate sqrt(ln(x)) for x = 1, 2, 3.
| x | u = ln(x) | y = sqrt(u) = sqrt(ln(x)) |
| 1 | 0 | 0 |
| 2 | .69 | .83 |
| 3 | 1.1 | 1.05 |
The first thing evaluated is ln(x) so let u = ln(x). We evaluate u then find sqrt(u) = sqrt(ln(x)).
How do we write this function y = sqrt(ln(x)) as a composite in the form f(g(x))?
The first thing that happens to x in sqrt(ln(x)) is ln(x).
The first thing that happens to x in f(g(x)) is g(x).
So g(x) = ln(x).
The next thing that happens in sqrt(ln(x)) is that we take the square root of the previous result.
The next thing that happens in f(g(x)) is that f is applied to the result.
So f is the square root function. Using 'dummy' variable z we write this as f(z) = sqrt(z).
Express y = (sqrt(x))^2 - 2 sqrt(x) + 3 as a composite f(g(x)).
If we evaluate u = sqrt(x) first then we can evaluate y = u^2 - 2 u + 3.
The first thing that happens to x is then g(x) = sqrt(x).
The next thing that happens is f(z) = z^2 - 2 z + 3.
This gives us
f(g(x)) = g(x)^2 - 2 g(x) + 3 = (sqrt(x))^2 - 2 sqrt(x) + 3.