Math 1122 Lecture 3 Notes

Math 1122 Lecture 3 Notes

Review: Function Composition

• Definition of Function:

  • Think of a function as a machine in a box that takes input, processes it, and produces output.

• Example Function:

  • Let f(x) = 2x.

Combining Functions

• Function Composition:

  • The notation for composing functions is written as fog, which means to apply g first and then apply f.

  • Composition is read right-to-left:

  • $f(g(x))$

• Example of Composition:

  • Let g(x) = x + 3.

  • Therefore, $f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6$.

Step-by-Step Examples of Composition

• Example 1:

  • $2(g(-2)) =$

  • Applying g: $g(-2) = -2 + 3 = 1$

  • Therefore, $2(g(-2)) = 2(1) = 2$.

• Example 2:

  • For $g(6)$:

  • $g(6) = 6 + 3 = 9$.

  • Therefore, $f(g(6)) = f(9) = 2(9) = 18$.

Consolidating Compositions

• Function compositions can be simplified or combined.

  • Example:

  • From $g(-1)$:

  • Find $g(-1) = -1 + 3 = 2$

    • Then, $f(g(-1)) = f(2) = 2(2) = 4$.

Identifying Patterns

• Notice that in function composition, certain functional patterns repeat irrespective of prior or subsequent functions.

  • For a constant input-output relationship, the result will always remain consistent across compositions.

Graph Transformations

• Impact of Composition on Graphs:

  • How function compositions alter graphs can sometimes be inferred directly.

• Example Transformation:

  • If $g(x) = 2f(x)$, there is a vertical stretch by a factor of 2.

  • This occurs because all output values are doubled.

• Further Transformation:

  • If $h(x) = 2f(x) - 1$, this means the graph is stretched by a factor of 2 vertically and then shifted downwards by 1.

Order of Operations in Transformations

• Graph transformations follow order-of-operations rules:

  • Operations on output values (after function) happen post-evaluation.

• Example, if $f(x) = x$ and $h(x) = 2f(-x)$, transformations affect the output as follows:

  • Reflect across the y-axis, and then stretch vertically by 2.

Transformation Before and After

• Quick example with point evaluation:

  • Consider point $F(1) = 4$.

  • Then, point transformation impacts will differ for $g(x) = -2f(x) + 5$.

  • Achieve outputs such as:

  • $g(1) = -2 imes f(1) + 5 = -2 imes 4 + 5 = -3$.

Evaluating h(x) = f(-2x + 5) Example

• Important considerations arise when evaluating compositions like $h(x) = f(-2x + 5)$:

  • The inner function operations dictate output. First, evaluate:

  • Solve for input: $-2x + 5 = 1$ and deduce $x = 2$.

  • Calculate the output: $h(2) = 4$.

Understanding Input and Output Relationships

• Remember that:

  • Plugging in values versus solving equations represent opposite operations within functions!

  • For instance, given $f(x) = y$:

  • Plugging in $x$ yields $y$.

  • Conversely, solving $y$ for $x$ requires reversing this operation.

Shifting and Stretching Rules

• Horizontal Shifts:

  • $f(x + k)$ shifts left by k units, while $f(x - k)$ shifts right, affecting x-values of graphs.

• Stretching and Shrinking:

  • The effects of a scalar modification before an input operation can precondition results, such as with $f(2x + 3)$.

Capstone Example with Quadratics

• Consider the function $f(x) = x^2$.

  • The transformations could be articulated by compositions like $g(x) = 3\big(- rac{1}{2}x + 5\big)^2 - 3$:

  • Analysis reveals a sequence of transformations:

    • Shift left by 5, flip horizontally, stretch horizontally by 2, stretch vertically by 3, and shift down by 3.