Function Notation

Linear Equations

$f(x)=4x$

It is possible to prove that for f(x):

$f(x+y)=f(x)+f(y)$

$4(x+y)=4x+4y$

However we can see that this not true for g(x), which is a linear equation with a small shift:

$g(x)=4x+2$

$g(x+y) \neq g(x)+g(y)$

$4(x+y) \neq 4x+2 +4y+2 \newline 4x+4y \neq 4x+4y+4$

Inverse Equations

An inverse equation has different properties to a linear equation

$f(x)=\frac{1}{x}$

$f(xy) \neq f(x)+f(y)$

Since,

$\frac{1}{xy} \neq \frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}$

The left hand side needs to be multiplied by (x+y) in order to equal the right hand side therefore:

$(x+y) f(xy)=f(x)+f(y)$

Exponential and Logarithmic Functions

Logarithmic functions have the characteristic that adding two logs together is the same as multiplication of the variables:

$f(x)+f(y)=log(x)+log(y)=log(x+y)=f(xy)$

$f(x)-f(y)=log(x)-log(y)=log(\frac{x}{y})=f(\frac{x}{y})$

Exponential functions have the characteristic that multiplying to exponentials together is the same as addition for the variables:

$f(x+y)=e^{x+y}=e^{x}e^{y}=f(x)f(y)$

Example

2013 exam 2

In order to answer this question we need to substitute the notation from the questions and see which works

A) $\frac{4x^{2}}{2}-2\frac{x^{2}}{2} \neq 0$
B) $\sqrt{4x}-2\sqrt{2x} \neq 0$
C) $4x-4x=0$
D) $log_{e}(x)-2log_{e}(\frac{x}{2}) \neq 0$
E) $2x-2-(2x_4) \neq 0$

Only C works and that is our answer

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