Overview
This lecture explains how to solve standard function questions, including evaluating functions at a given value and solving for input values, both algebraically and using graphs.
Understanding Functions and Notation
- f(x) means "the value of the function f at x", and is often the same as y.
- The expression f(x) = y represents the point (x, y) in the coordinate system.
- The variable inside parentheses (e.g., x in f(x)) is the input; changing it changes the function's output (y).
Algebraic Function Questions
- Given a function, e.g., f(x) = 2x - 3, to find f(2), substitute x = 2: f(2) = 2*2 - 3 = 1.
- To solve f(x) = 2, set 2x - 3 = 2 and solve for x:
- 2x - 3 = 2
- Add 3 to both sides: 2x = 5
- Divide by 2: x = 5/2 or 2.5
Function Names and Variables
- Different functions can have names like f, g, or s (e.g., g(x), s(t)), and the input variable can vary.
- The output depends on the input variable used (e.g., s(t) depends on t).
Graphical Function Questions
- If given a graph where y = f(x), to find f(2), locate x = 2 on the x-axis and read the corresponding y value.
- For example, if the graph shows y = -2 when x = 2, then f(2) = -2.
- To solve f(x) = 2 using the graph, find all x-values where y = 2.
- If the graph crosses y = 2 at x = -2 and x = 4, the solutions are x = -2 and x = 4.
Key Terms & Definitions
- Function (f(x)) — A rule assigning each input x exactly one output y.
- Evaluate — Find the output of a function for a specific input.
- Solve — Find input values that produce a specific output.
- Graph of a function — A visual representation of all (x, f(x)) pairs.
Action Items / Next Steps
- Practice solving function questions both algebraically and using graphs.
- Review homework or textbook problems related to function evaluation and solving equations with functions.