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# Core 3
## Functions
A function is a one to one mapping
### Domain and range of a function
The domain of a function is the set of valid input values
A function is defined by both a rule and a domain
### Inverse of a function
An inverse function is such that $ff^{-1} (x) = f^{-1} f(x) = x$
For a function to have a valid inverse function, it must be a one to one mapping
## Transformations and the modulus function
### Translations
Translating the function $y = f(x)$ by the vector $\begin{bmatrix} a \\ b \end{bmatrix}$ gives $y = f(x-a) +b$
$y = kf(x)$ is a stretch of $y = f(x)$ by a factor of $k$ in the $y$ direction
$y = f\left(\frac x k \right)$ is a stretch y a factor of $k$ in the $y$ direction
Reflecting $y=f(x)$ in the y axis gives $y=f(-x)$
The modulus of a function is given by:
$$|f(x)| = \left\{
\begin{array}{ll}
f(x) & \quad f(x) \geq 0 \\
-f(x) & \quad f(x) < 0
\end{array}
\right.
$$
## Inverse trigonometric functions
$\operatorname{asin}(x)$ has domain $-1 \leq x \leq 1$ and range of $-\frac{\pi}{2} \leq \operatorname{asin}(x) \leq \frac{\pi}{2}$
$\operatorname{acos}(x)$ has domain $-1 \leq x \leq 1$ and range of $0\leq \operatorname{acos}(x) \leq \frac{\pi}{2}$
$\operatorname{atan}(x)$ has domain $x \in \mathbb{R}$ and range of $-\frac{\pi}{2} \leq \operatorname{atan}(x) \leq \frac{\pi}{2}$
## Trig identities
$$\begin{align}
&\sin(x)^2 + \cos(x)^2 = 1 \\
&1 + \cot(x)^2 = \operatorname{cosec}(x)^2 \\
&1 + \tan(x)^2 = \sec(x)^2\end{align}$$
## Calculus
### Log rules
$$\begin{align}
&\ln(a) + \ln(b) = \ln(ab) \\
&\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \\
&n\ln(a) = \ln(a^n)\\
&\log_a b = \frac{\log_c(a)}{\log_c(b)}
\end{align}$$
### Differentials and integrals
$$\begin{alignat*}{3}
&\frac{d}{dx} e^x = e^x \quad &&\int e^x dx = e^x + C \\
&\frac{d}{dx} e^{ax + b} = ae^{ax+b} \quad &&\int e^{ax+b} dx = \frac{1}{a} e^{ax+b} + C\\
&\frac{d}{dx} e^{f(x)} = f'(x)e^{f(x)} \quad &&\int e^{f(x)} dx = \frac{1}{f'(x)} e^{f(x)} + C \\
&\frac{d}{dx} \ln(x) = \frac{1}{x} \quad &&\int \frac{1}{x} dx = \ln(x) + C \\
&\frac{d}{dx} \sin(x) = cos(x) \quad &&\int \sin(x) = dx = -\cos(x) + C \\
&\frac{d}{dx} cos(x) = -\sin(x) \quad &&\int \cos(x) dx = \sin(x) + C\end{alignat*}$$
### The chain rule
$$\frac{dy}{dx}\ = \frac{dy}{du} \frac{du}{dx}$$
### The product rule
$$ \frac{d}{dx} f(x) g(x) = f(x)g'(x) + f'(x)g(x) $$
or
$$\frac{d}{dx} uv = uv' + vu'$$
### The quotient rule
$$\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}$$
or
$$\frac{d}{dx} \frac{u}{v} = \frac{vu' - uv'}{v^2}$$
### Integration by inspection and substitution
$$\begin{align} &\int \cos(ax + b) dx = \frac{1}{a} \sin(ax + b) + C \\
&\int \sin(ax + b) dx = -\frac{1}{a} \cos(ax + b) + C \\
&\int \sec(ax + b)^2 dx = \frac{1}{a}\tan(ax + b) + C \\
&\int \frac{f'(x)}{f(x)} dx = ln|f(x)| + C \end{align}$$
### Integration by parts and standard integrals
$$\int f(x) \frac{d}{dx} g(x)\ dx = f(x)g(x) - \int g(x) \frac{d}{dx} f(x) \ dx$$
More simply
$$\int u \frac{dv}{dx} \ dx = uv - \int v \frac{du}{dx} \ dx$$
In general it is easier to make $u$ the simpler part of the given integrand.
### Volume of a revolution
The volume of a function $y = f(x)$ rotated fully around the $x$ axis is given by
$$V = \int _a ^b \pi y^2 \ dx$$
To find the volume of a revolution around the $y$ axis, rearrange the function to the form $x = g(y)$.
## Numerical solutions and iterative methods
### Iteration
Iteration requires a formula in the form $X_{n+1} = f(x_n)$
An iterative formula converges if $|f'(x)| < 1$ | for certain values of x
If the iteration converges to a limit the limit can be found by setting $x_{n+1} = x_n = L$
### Trapezium rule
$$\int _a ^b y\ dx \approx \frac{b-a}{n} [y_0 + y_n + 2(y_1 + ... + y_{n-1})]$$
If $y$ tends upwards the rule will overestimate, and if $y$ tends downwards the rule will underestimate.
### Mid-Ordinate rule
$$\int _a ^b y\ dx \approx \frac{b-a}{n} (y_{\frac{1}{2}} + y_{\frac{3}{2}} + ... + y_{n - \frac{3}{2}} + y_{n - \frac{1}{2}}) $$
Where $n$ is even.
### Simpson's rule
$$\int _a ^b y\ dx \approx \frac{b-a}{3n} [y_0 + y_n + 4(y_1 + y_3 + ... + y_{n-1}) + 2(y_2 + y_4 + ... + y_{n-2})]$$
### Cobweb diagram
Plot the two functions
A cobweb diagram is plotted by drawing up from the first x value, before rotating around, alternating between each of the functions for each value of x.
This is more appropriate when the values converge around the root rather than towards it.
### Staircase diagram
Plot the two functions
A staircase is plotted by drawing a line up from each x value to one of the functions, and then drawing across to the next intersect with the other function.
