Evaluating Functions: Step-by-Step Guide
A function is a rule that takes an input and gives back one output. We write f(x) to name a function, and it is read 'f of x'. To evaluate a function at a number, like f(3), replace every x in the rule with that number and then work out the answer.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Find the rule and the input. Read the rule for f(x). Then find the number you are asked to use, like the -3 in f(-3).\[f(x) = \text{rule},\quad \text{find } f(a)\]
- Replace x with the input. Replace every x in the rule with the input number. Put the number in parentheses.\[f(a) = \text{rule with } x \to (a)\]
- Work it out with PEMDAS. Follow the order of operations to find the answer.\[f(a) = \text{simplified value}\]
- Write the answer, or the point if needed. If you are making a table or graph, write the input and output together as a point (input, output).\[(a,\; f(a))\]
Worked example
Given f(x) = 2x^2 - 3x + 1, find f(-3).
Step 1: The rule is f(x) = 2x^2 - 3x + 1. We want f(-3).
Step 2: Replace x with -3: f(-3) = 2(-3)^2 - 3(-3) + 1.
Step 3: Do the exponent first: (-3)^2 = 9. Now we have 2(9) - 3(-3) + 1.
Step 4: Multiply: 2 x 9 = 18, and -3 x -3 = 9. Now we have 18 + 9 + 1.
Step 5: Add: 18 + 9 + 1 = 28. So f(-3) = 28.
Check: Go through each part again. 2(9) = 18, -3(-3) = 9, and then + 1. The total is 28.
Tips and common mistakes
- Always put the input in parentheses when you replace x. This keeps negative signs from causing trouble.
- Replace every x with the same number.
- f(x) means 'f of x'. It does not mean f times x.
- Be careful: f(2) + f(3) is usually not the same as f(2 + 3).
- After you replace x, follow PEMDAS carefully.
- Some functions do not work for every input. You cannot divide by 0 in 1/x, and you cannot take the square root of a negative number.
- For a piecewise function (one with different rules for different inputs), first find which rule matches your input. Then use only that rule.