Function notation is crucial for understanding equations, graphs, parabolas, and polynomials on the SAT. Functions are typically denoted as \( f(x) \) (pronounced "f of x") or \( g(x) \), but they can also use any variables, such as \( g(t) \), \( h(z) \), or \( m(r) \).
Generally, you can treat \( f(x) \) as \( y \). For example, \( f(x) = 2x + 3 \) is equivalent to \( y = 2x + 3 \).
When given a function \( f(x) \), substitute the term inside the parentheses for every occurrence of \( x \). For instance, given:
\[ f(x) = 2x^2 + 4x \]
To find the value of \( f(2) \):
\[ f(2) = 2(2)^2 + 4(2) \]
\[ f(2) = 2(4) + 4(2) \]
\[ f(2) = 8 + 8 \]
\[ f(2) = 16 \]
Simply plug the number in the parentheses into the variable. Easy peasy!
A composite function is essentially a function within a function. There are numerous ways the SAT might test this concept, but understanding it's simplicity can earn you points. Let's look at an example:
If \( f(x) = 3x + 7 \) and \( g(x) = 2x + 9 \), find \( f(g(x)) \).
The process is the same as with numbers: substitute the expression inside the parentheses for the variable.
\[ f(g(x)) = 3(g(x)) + 7 \]
Now substitute \( g(x) \) with its expression:
\[ f(g(x)) = 3(2x + 9) + 7 \]
\[ f(g(x)) = 6x + 27 + 7 \]
\[ f(g(x)) = 6x + 34 \]
Let's explore another example.
If \( f(x) = 2x^2 - 3x + 7 \) and \( g(x) = 2x + 4 \), find the value of \( f(g(2)) \).
First, break the problem into two parts. Notice, they are not asking for \( f(g(x)) \) or just \( g(2) \), but both at the same time. So, start with the inside and work outwards.
First, find \( g(2) \):
\[ g(2) = 2(2) + 4 \]
\[ g(2) = 4 + 4 \]
\[ g(2) = 8 \]
Now find \( f(g(2)) \). Since \( g(2) = 8 \), substitute 8 into \( f(x) \):
\[ f(g(2)) = f(8) \]
\[ f(8) = 2(8)^2 - 3(8) + 7 \]
\[ f(8) = 128 - 24 + 7 \]
\[ f(8) = 111 \]
So \( f(g(2)) = 111\)
On the SAT, you might be given a graph along with its function. Typically, you will be asked about transformations and properties like the x-intercept and y-intercept.
All you need to do is memorize two rules. I’ll use \( c \) as a constant to denote a number.
• If given \( f(x) \pm c \), the function shifts up or down (adding \( c \) moves it up, subtracting \( c \) moves it down).
• If given \( f(x \pm c) \), the function shifts left or right (adding \( c \) moves it left, subtracting \( c \) moves it right).
In simpler terms: Outside the parentheses is up and down; inside the parentheses is left and right.
Let's look at some graphs to see this concept in action.
This graph shows \( f(x) \) and \( f(x) + a \). Here, \( a \) is a number that you can adjust using the slider. When you increase \( a \), the graph moves up. When you decrease \( a \), the graph moves down. Try it out to see how the vertical position of the graph changes.
This graph shows \( f(x) \) and \( f(x + a) \). In this case, \( a \) affects the horizontal position of the graph. When you increase \( a \), the graph shifts to the left. When you decrease \( a \), the graph shifts to the right. Use the slider to see these horizontal shifts in action.
Feel free to use Desmos or any graphing tool to experiment with transforming functions. Practice is the best way to understand and memorize these concepts.
For more on composite functions
Extended video lesson on functions
For more on graph transformations