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).
Suppose you're asked to solve the equation $\,x^2 = -1\,.$
You'll need some clarification:
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.
Notice something?
| 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 $\,f(x) = x^2 + 1\,.$ |
| 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?
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On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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