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In this lesson, you will learn about functions: rules which assign outputs to inputs.
A function is a rule which assigns a single output ($y$-value) to each of its inputs ($x$-values). Not every number needs to be a possible input.
The table below gives an example of a function. The $(x,y)$ values shown in the table are also plotted on the grid to the left.
Click . Now the $(x,y)$ values shown in the next table below are plotted on the grid to the left.
In table-qn, you looked at a function where the rule was given by a table. You can also use a graph to give the rule for a function. The graph plotted on the grid to the left is an example of this.
You can click inside the grid and drag the vertical bar around to see the $y$-value corresponding to each $x$-value, under the words “Trace at:” to the right of the grid. For example, you can set the vertical bar to $x=2$. Then the information to the right of the grid will tell you that the function assigns the output 4 to the input 2.
Click to see another graph plotted on the grid to the left.
Another way to give a rule is to use a formula. For example, there is a function that takes each input $x$ and assigns to it the output $2x-3$. If you give it the input $1$, it produces the output $2(1)-3=-1$.
Each row of the table below gives an input to this function. Enter the corresponding output.
Click to see the entire graph of this function. This is the graph of $y=2x-3$, so it is a straight line.
The function which assigns the output $-x+2$ to the input $x$ is graphed on the grid to the left.
A function whose output always increases as its input increases is called an increasing function. A function whose output always decreases as its input increases is called a decreasing function.
A function whose graph is a straight line (that is, a function given by a formula of the form $mx+b$) is called a linear function.
In general:
A function is increasing whenever its graph goes up and to the right. It is decreasing whenever its graph goes down and to the right.
A linear function is increasing whenever the slope of its graph is positive. It is decreasing whenever the slope of its graph is negative.
A ball is dropped from a window 30 meters above the ground. We will study the function whose input is the number of seconds $t$ since the ball was dropped, and whose output is the height of the ball $\cl"red"h$ above the ground (in meters). This function is graphed on the grid to the left.
A catapult shoots a pumpkin into the air. The function whose input is the number of seconds $t$ since the pumpkin was released, and whose output is the height of the pumpkin $\cl"red"h$ above the ground in meters, is graphed on the grid to the left.