Functions
Function notation f(x), evaluating functions, input/output tables, domain, and range — does NOT include graphing
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Have you ever used a vending machine? You push a button (the input), and out pops a snack (the output). In math, a function is just like a magical smoothie machine! 🍓 You drop a number in, the machine chops it up, mixes it with some math rules, and pours out a brand new number.
When you see something like , don't let those letters scare you! The just stands for 'function', and the is the ingredient you drop inside. If you drop in a (we write that as ), the machine multiplies it by and adds to give you . Delicious! 🥤
Graphing is just a way to take a picture of what our machine is doing. We use a coordinate plane, which looks like a giant piece of grid paper. Every time we put an into our machine and get a out, we get a point (x, y). If we plot a bunch of points and connect them, we get a line!
On the ISEE, you'll see questions asking you to find points on a line, figure out the slope (how steep the line is, like a skateboard ramp! 🛹), or compare functions in the Quantitative Reasoning section. The great news? The ISEE has no penalty for guessing! So, always pick an answer, even if the math machine seems a little tricky that day. Grab your pencil and let's make some math smoothies!
Practice Questions
4 practice questions for ISEE Upper Level
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- To find the value of , substitute and into the given function rule. First, square : . Next, subtract : . Finally, divide by 2: .
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- According to Newton's Law of Cooling, the temperature of a hot object decreases rapidly at first when the temperature difference between the object and its surroundings is large. As the tea cools and approaches the room temperature (70°F), the rate of cooling slows down, causing the graph to level off asymptotically at 70°F. This creates an exponential decay curve.
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- First, evaluate each function at the given values.
Now, order these values from least to greatest: . Substituting the function notation back in gives .
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- To evaluate the composite function , work from the inside out. First, find by substituting 4 for : . Next, substitute this result back into the function to find : .
Tips & Strategies
- Rise over Run! When finding slope on a graph, always count how far UP or DOWN you go (rise) before you count how far LEFT or RIGHT you go (run). Putting the on top of your fraction is a super common trap!
- For Quantitative Comparison questions, don't solve for or if you don't have to! If they just ask for the y-intercept, grab the from and compare. Save your brainpower for harder questions!
- Use the answers! If you forget how to graph a line, you can test the points given in the answer choices by plugging the and values back into the equation to see if it makes a true math sentence.
Common Mistakes
- Watch out for negative signs when finding the slope! If you are subtracting a negative number, like , remember that two negatives make a positive, so it becomes .
- Don't get tricked by . It does NOT mean times ! It is just the name of the function, like saying 'Machine A'.
Frequently Asked Questions
What does f(x) actually mean?
It's just a fancy math way of writing ! It tells us that the output (the answer) depends on the input (the ). You say it out loud as 'f of x'.
Do I need to memorize the slope formula for the ISEE?
Yes! The ISEE doesn't give you a formula sheet. Memorize and remember to always put the values on top!
What if I get totally stuck on a function question?
Since the ISEE has no guessing penalty, never leave a question blank! If you're stuck, try plugging in simple numbers like or for to see what happens, or just make your best guess and move on.