General Solutions where f(x) = k (Constant Types)

Overview

When your function, f(x), in an equation is equal to a constant, k, you’re dealing with a constant type function. This might seem simple, but it’s crucial to understand how this affects your solutions.

Finding Solutions

In an equation of the form f(x) = k, your primary aim is to find values of x that make the equation true.
Different techniques are needed depending on whether f(x) is a polynomial, a trigonometric function, a logarithm, etc.

Polynomial Functions

For a polynomial function, set the polynomial equal to k and solve for x. This might involve factoring, completing the square, or using the quadratic formula.
Every polynomial equation of degree n has n roots (solutions for x), although some of these might be complex (involving the square root of a negative number).

Trigonometric Functions

For a trigonometric function, use known values (such as sin(30°) = 0.5 or cos(0) = 1) where possible.
Use the CAST rule to find additional solutions. CAST stands for “Cosine positive in the Fourth and First quadrants, All positive in the first quadrant, Sine positive in the first and second quadrants, and Tan positive in the third and fourth quadrants”.
To find a general solution, consider the periodicity of trigonometric functions. For example, since sin(x) and cos(x) repeat every 360° (or 2π radians), if x = a is a solution then so is x = a + 360n or x = a + 2πn for any integer n.

Logarithmic and Exponential Functions

For a logarithmic function of the form log_a(f(x)) = k, remember that this is equivalent to the equation a^k = f(x).
For an exponential function of the form a^f(x) = k, take the logarithm of both sides to solve for x.
Remember the properties of logarithms and exponentials, such as log_a(a^x) = x and a^log_a(x) = x.
Also consider the domain of these functions: a logarithm function is only defined for x > 0, while an exponential function is defined for all x.

Understanding this concept is crucial in making progress in the Core Pure component of the curriculum. You will find these types of functions utilized in numerous applications.