Function Scaling and Combination Study Guide

Principles of Function Scaling

• Definition of Scaling: Scaling a function means multiplying the entire function by a real number coefficient.

• Method of Distribution: To scale a function by a number, distribute the multiplier to every term within the function definition.

• General Model Example:

$f(x) = 3x + 4$

$-2f(x) = -2(3x + 4)$

$-2f(x) = -6x - 8$

Base Functions for Evaluation

• The following individual functions are defined for algebraic scaling and combination operations:

$f(x) = 2x + 5$

$g(x) = x - 9$

$h(x) = -5x + 12$

Single-Function Scaling Problems

• Problem 10: Scale $f(x)$ by a factor of $5$

$5f(x) = 5(2x + 5)$

$5f(x) = 10x + 25$

• Problem 11: Scale $g(x)$ by a factor of $-3$

$-3g(x) = -3(x - 9)$

$-3g(x) = -3x + 27$

• Problem 12: Scale $h(x)$ by a factor of $4$

$4h(x) = 4(-5x + 12)$

$4h(x) = -20x + 48$

Order of Operations for Scaling and Combining Functions

• Procedural Rule: When an expression requires both scaling functions and combining functions, the operations must be executed in a specific sequence:

  1. Perform the scaling step first by multiplying each scalar constant across its respective function.

  2. Combine like terms second to simplify the expression into its final form.

• Demonstration Example: Combine $3f(x) + 2g(x)$

$3f(x) + 2g(x) = 3(2x + 5) + 2(x - 9)$

$3f(x) + 2g(x) = 6x + 15 + 2x - 18$

$3f(x) + 2g(x) = 8x - 3$

Multi-Function Scaling and Combination Problems

• Problem 13: Calculate $3f(x) + 2h(x)$

$3f(x) + 2h(x) = 3(2x + 5) + 2(-5x + 12)$

$3f(x) + 2h(x) = 6x + 15 - 10x + 24$

$3f(x) + 2h(x) = -4x + 39$

• Problem 14: Calculate $g(x) - 3h(x)$

$g(x) - 3h(x) = (x - 9) - 3(-5x + 12)$

$g(x) - 3h(x) = x - 9 + 15x - 36$

$g(x) - 3h(x) = 16x - 45$

• Problem 15: Calculate $2h(x) + 5f(x)$

$2h(x) + 5f(x) = 2(-5x + 12) + 5(2x + 5)$

$2h(x) + 5f(x) = -10x + 24 + 10x + 25$

$2h(x) + 5f(x) = 49$