Section 1.3 New Functions from Old

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

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

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

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

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

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

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)

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

Graphs of Inverses Consider the function

The inverse is

Plot of the two graphs together

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

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.