02Ifunctioneqns

toc = Functional Equations =

Most equations are written in terms of unknown variables such as x. In this section, we look at equations that are written in terms of unknown functions such as f(x).

The most common question will be to verify that a given f(x) satisfies the equation. Generally the technique is to substitute the function into the equation and simplify.

If you end up with something which can never happen (like 2 = 3) then f(x) does **not** satisfy the equation. If you end up with something which is always true (like 5 = 5) then f(x) **does** satisfy the equation. If you end up with values for x (such as x = ±3) then f(x) only satisfies the equation for those values.

** Example 1 **
Given f(x) = 2x + 3, determine if f(x) satisfies the equation 3f(x) = f(3x)

__**Solution:**__

3f(x) = 3(2x + 3) f(3x) = 2(3x) + 3

Equation is: 3f(x) = f(3x) 3(2x + 3) = 2(3x) + 3 6x + 9 = 6x + 3 9 = 3

This is __never__ true so f(x) does __**not**__ satisfy the equation

** Example 2 **
Given f(x) = 5x, determine if f(x) satisfies the equation 3f(x) = f(3x)

__**Solution:**__

3f(x) = 3(5x) f(3x) = 5(3x)

Equation is: 3f(x) = f(3x) 3(5x) = 5(3x) 15x = 15x

This is true for all values of x, so f(x) **__does__** satisfy the equation

Functional Equations on the Classpad
The CAS calculator can verify functional equations.

First enter the function by typing: Define f(x) = 2x + 3 Use the virtual keyboard ABC tab for the name of the function (f) and use the variable keypad for **x**
 * Define** is in the ACTION menu, COMMAND submenu

Then give the command to solve the equation by typing Solve (3f(x)=f(3x)) Use the virtual keyboard ABC tab for the name of the function (f) and use the variable keypad for **x**
 * Solve** is in the ACTION menu, EQUATION submenu

As you can see from the screen capture, the calculator will respond with {** No Solution **}

In a second example on the same screen, I have used the function f(x) = 5x. The calculator has responded with {** x=x **}, which is always true, so the function f(x) = 5x **does** satisfy the equation. Go to top of page: flat

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