Exam Questions - General solutions where f(x) = kx (linear types)

Linear Types - f(x) = kx

This type of equation represents linear functions, where f(x) = kx with k being a constant.
A general solution to an equation involving such functions must account for all possible values of x that satisfy the equation.
These types of questions often involve identifying values that satisfy the equation, interpreting and manipulating the equation, and plotting or determining characteristics of the graph of the function.

To solve for x in such equations, use basic algebraic techniques like isolating x, factoring, or expanding expressions. Remember that x can be positive, negative or zero.
Whenever k ≠ 0, the equation f(x) = kx always has a unique solution.
When k = 0, all values of x are solutions. This is because kx = 0 for all x, so f(x) = 0 for all x.
When calculating a solution, check your answer by substituting it back into the original equation to ensure it satisfies the equation.

The graph of f(x) = kx is a straight line passing through the origin (0,0) with a slope of k.
The line is upward sloping if k is positive, downward sloping if k is negative, and horizontal if k = 0.
Be able to sketch these lines and understand how the value of k affects the slope and hence the shape of the line.

This linear function might be used to model real-world scenarios, such as constant speed or a regular rate of change.
In these problems, identify the key variables and their relationships, set up the correct f(x) = kx equation, solve for x or k, and interpret the solution in the context of the problem.
Always validate your solution within the context of the problem - does it make sense and fit within the given parameters?

It’s important to remember that f(x) = kx solutions are not just meant for finding a singular answer, but rather a representation of a set of all possible solutions.
Be familiar with different forms of expressing the solutions: as single values, as intervals of values or as infinity in cases where there are endless solutions.
These equations can often lead to further exploration in mathematical areas such as sequences, functions and calculus.