Functions and Evaluation of Composite Functions

Functions and Their Evaluations

Function Definitions

• Function notation is denoted by $y = f(x)$, where:
  • $y$ represents the output of the function.
  • $f$ denotes the function itself.
  • $x$ is the input variable.

Evaluating Composite Functions

• To evaluate a composite function such as $g(f(0))$, follow these steps:
  1. First, evaluate the inner function $f(0)$:
     • Find the corresponding output value when $x = 0$ for the function $f(x)$.
  2. Next, take the result from step 1 and use it as the input for the outer function $g(x)$.
     • Thus, evaluate $g(f(0))$ using the result from step 1 as input for $g(x)$.

Graphical Interpretation

• The graphs of the functions $y = f(x)$ and $y = g(x)$ can be visualized in a coordinate plane:
  • y-axis: Coordinate values range from -10 to 10.
  • x-axis: Coordinate values are defined from -10 to 10.
• Example outputs based on different input values can be extracted from the graphical representations.

Example Evaluation Calculation

• Assume a specific example where $f(0) = k$, where $k$ is the y-coordinate of the function $f(x)$ at point $x = 0$.
  • Once this has been calculated, we can substitute $k$ into the function $g(k)$ to get the final output value.
• If the values from the graph of $g$ suggest that for an input of $k$, the output is $m$, then:
  • Final answer: $g(f(0)) = m$.

Specific Example Output Values

• Consider midpoints of the graph:
  • If $f(0) = 8$, and assuming from the graphs that $g(8) = -7$, then:
    • $g(f(0)) = g(8) = -7.