Function Notation: Unveiling the Mystery

Hello Math Students! In our 10-11-2013 class, we dove into the world of Function Notation. Function notation is a way to represent a function using the symbol $f(x)$ rather than $y$. Think of it as a different way of writing equations, where $f(x)$ is read as "f of x" and it represents the value of the function at a specific input $x$. It's simply expressing a function in terms of "x". Let's break it down:

Understanding the Basics

  • $f(x)$ represents the output of the function for a given input $x$.
  • It's just a different way to write $y$. So, $y = 2x + 3$ can also be written as $f(x) = 2x + 3$.

Evaluating Functions

The real power of function notation comes when we want to evaluate the function for specific values. For example, if we have the function $f(x) = 2x + 3$, we can find $f(3)$ by substituting $x$ with $3$:

$$f(3) = 2(3) + 3 = 6 + 3 = 9$$

Similarly, to find $f(-4)$:

$$f(-4) = 2(-4) + 3 = -8 + 3 = -5$$

We can also substitute variables with other variables or expressions. For instance, to find $f(a)$:

$$f(a) = 2(a) + 3$$

And to find $f(x-2)$:

$$f(x-2) = 2(x-2) + 3 = 2x - 4 + 3 = 2x - 1$$

More Examples

Let's look at some more examples to solidify our understanding. Consider the function $f(x) = -2x + 7$. Then:

  • $f(2) = -2(2) + 7 = -4 + 7 = 3$
  • $f(-3) = -2(-3) + 7 = 6 + 7 = 13$
  • $f(m) = -2m + 7$
  • $f(x+2) = -2(x+2) + 7 = -2x - 4 + 7 = -2x + 3$

Functions can also involve exponents. If $f(x) = x^2 + x - 2$, then:

  • $f(2) = (2)^2 + (2) - 2 = 4 + 2 - 2 = 4$
  • $f(-3) = (-3)^2 + (-3) - 2 = 9 - 3 - 2 = 4$
  • $f(m) = m^2 + m - 2$

And even absolute values! If $f(x) = |2x - 5| + 2$, then:

  • $f(-2) = |2(-2) - 5| + 2 = |-4 - 5| + 2 = |-9| + 2 = 9 + 2 = 11$
  • $f(x+1) = |2(x+1) - 5| + 2 = |2x + 2 - 5| + 2 = |2x - 3| + 2$

Homework

To practice, please complete the following problems:

  • Page 72: #43-50
  • Page 72: #53, 54

Keep practicing, and you'll become a function notation pro in no time!