Solve an Equation? Find a Zero? Your Choice!
In this section, ‘equation’ refers to an equation in one variable, and ‘function’ refers to a function that takes a single input (a number) and gives a single output (a number).
Solve an Equation ...
Suppose you're asked to solve the equation: $$x^2 = -1$$
You'll need some clarification:
- Do you want only real-number solutions? If so, there aren't any—no real number, when squared, equals negative one.
- Are you allowing any complex number to be a solution? Then, there are two solutions, $\,i\,$ and $\,-i\,.$
Find a Zero ...
Now, suppose you're asked to find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$
- That is, you want inputs whose output is zero.
- That is, you want values of $\,x\,$ for which $\,f(x) = 0\,.$
- That is, you want values of $\,x\,$ for which $\,x^2 + 1 = 0\,.$
- That is, you want values of $\,x\,$ for which $\,x^2 = -1\,.$
Now, you'll need the same clarification as above.
Your Choice!
Notice something?
Every request to solve an equation can be rephrased as a request to find the zeros of a function, as follows:
- Get an equivalent equation with zero on the right-hand side.
- Define the function using the left-hand side of the equation.
For example:
Initial request: | Solve the equation $\,x^2 = -1\,.$ |
Get an equivalent equation with zero on the right-hand side: | $x^2 + 1 = 0$ |
Define a function $\,f\,$ using the left-hand side of the equation: | $f(x) = x^2 + 1$ |
Rephrase the initial request: | Find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$ |
Every request to find the zeros of a function can be rephrased as a request to solve an equation, as follows:
- Set the output from the function equal to zero.
- Solve the resulting equation.
For example:
Initial request: | Find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$ |
Set $\,f(x)\,$ equal to zero: | $x^2 + 1 = 0$ |
Rephrase the initial request: | Solve the equation $\,x^2 + 1 = 0\,.$ |
In both cases, you'll need clarification. Are you wanting only real number solutions/zeros? Or, are you allowing any complex number for the solutions/zeros?