section 1.3 new functions from old. plot f(x) = x 2 – 3 and g(x) = x 2 – 6x + 1 on the same set...

Section 1.3New Functions from Old

• Plot f(x) = x2 – 3 and g(x) = x2 – 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(x2 – 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

4)( 2 xxf

• 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

xxxf 4)( 3

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)

52)(

2)( 2

ttg

xxxf

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

).(ifonlyandif)(1 tfQtQf

Graphs of Inverses• Consider the function ( ) 3f x x

• The inverse is 3)( 21 xxf

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

3)( 2 xxg

• 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.