1. Section 1.3 New Functions from Old

2. Plot f(x) = x 2 – 3 and g(x) = x 2 – 6x + 1 on the same set of axes –What is the relationship between the two graphs? If we rewrite g(x) as g(x) = (x – 3) 2 – 3 – 5 we can see it as f(x) being shifted. What is the shift? –3 units to the right and 5 units down What happens if we have another function, h(x) = -2(x 2 – 3)? –Vertical stretch by 2 units –Flipped over x-axis

3. In General f(x + a) is a shift a units to the left f(x – a) is a shift a units to the right f(x) + a is a shift a units up f(x) – a is a shift a units down

4. In General If we have a constant k such that y = k·f(x) then –If k > 1, then the graph of f is vertically stretched –If 0 < k < 1, then the graph of f is vertically compressed –If -1 < k < 0, then the graph of f is vertically compressed and reflected about the x-axis –If k < -1, then the graph of f is vertically stretched and reflected about the x-axis

5. In general for a function f(x) If y = f(kx) then –If k > 1 then we have a horizontal compression by a factor of 1/ k –If 0 < k < 1 then we have a horizontal stretch by a factor of 1/ k –If -1 < k < 0 then we have a horizontal stretch plus a reflection across the y-axis –If k < -1 then we have a horizontal compression plus a reflection across the y-axis

6. Plot the function What would this function look like if it were reflected over the y-axis? Find h(x) = f(-x) Since f(-x) = f(x) we have an even function which means it is symmetric about the y-axis

7. Plot the function Find h(x) = f(-x) Since f(-x) = -f(x) we have an odd function which means it is symmetric about the origin which is the same as reflected over both the x and y-axis

8. Compositions of Functions Using the following two functions: –Let’s find algebraic rules for h(x) = g(f(x)) and k(t) = f (g(t)) –Using your new functions find h(2), h(6), k(2) and k(6)

9. Inverse Functions Suppose Q = f(t) is a function with the property that each value of Q determines exactly one value of t. The f has an inverse function, f -1 and If a function has an inverse, it is said to be invertible

10. Graphs of Inverses Consider the function

11. The inverse is

12. Plot of the two graphs together

14. Horizontal Line Test A function must be one-to-one in order to have an inverse (that is a function) A function is one-to-one if it passes the horizontal line test –A horizontal line may hit a graph in at most one point We can restrict the domain of functions so their inverse exists –For example, if x ≥ 0, then we have an inverse for

15. Suppose that P(x) represents the total amount of profit that a company has earned in thousands of dollars as a function of how many items they have sold, x. Answer the following and be sure to include your units. –Interpret P(330) = 81.1 –Interpret P -1 (100) = 400.