Exhaustive Notes on Function Transformations: Vertical and Horizontal Shifts, Domain and Range Analysis

Cost Function Representation:
- The variable $x$ represents time or input quantity, and $y = f(x)$ represents the total cost associated with $x$ .
- At $x = 0$ , the cost is $\\n20$ . This value at zero represents the fixed cost (the initial investment required regardless of production output or time elapsed).
- At $x = 1$ , the corresponding cost is $\\n25$ .
- At $x = 2$ , the corresponding cost is $\\n50$ .
Adjustments for Inflation or Pricing Changes:
- External economic factors such as inflation require pricing adjustments, which translate mathematically to shifting the cost function.
- An upward adjustment of $\\n5$ changes the initial fixed cost at $x = 0$ from $\\n20$ to $\\n25$ , while preserving the structural behavior of the rest of the function.
- The updated function $g(x)$ is directly related to the original function $f(x)$ by adding a constant shift of $5$ units: $g(x) = f(x) + 5$
Definition of Vertical Shift:
- Adding a constant $k$ to a function $f(x)$ shifts the graph vertically by $k$ units upwards.
- Formal algebraic definition: $g(x) = f(x) + k$
- If $k > 0$ , the transformation is an upward vertical shift by $k$ units.
- If $k < 0$ , the transformation is a downward vertical shift by $|k|$ units.
- Example: For the function $g(x) = x^2 + 4x - 5$ , subtracting $5$ from the base function $f(x) = x^2 + 4x$ shifts the entire original function downward by $5$ units.

Domain Behavior Under Vertical Shifts:
- A vertical shift alters only the output values of a function, leaving the input values entirely unaffected.
- Consequently, the domain remains completely unchanged during a vertical shift.
Range Behavior Under Vertical Shifts:
- Because output values are directly incremented or decremented by the vertical shift constant $k$ , the range changes directly by $k$ units.
Concrete Example Analysis:
- Given a function with a domain that explicitly excludes $0$ : $\text{Domain} = [-3, 0) \cup (0, 3]$
- Range of the original function $f(x)$ : The minimum possible output is achieved when plugging in $x = -3$ or $x = 3$ , yielding a minimum output value of $1$ . Any input closer to $0$ yields an output strictly greater than $1$ . $\text{Original Range} = [1, \infty)$
- Effect of shifting $f(x)$ upward by $3$ units ( $k = 3$ ):
- New Domain: Remains $[-3, 0) \cup (0, 3]$ .
- New Range: The starting lower bound of the range shifts from $1$ to $1 + 3 = 4$ . Thus, the updated range is $[4, \infty)$ .

Definition and Directions:
- A transformation inside the argument of a function, written as $f(x - h)$ or $f(x + h)$ , represents a horizontal shift.
- Right Shift: $f(x - h)$ shifts the function $f(x)$ by $h$ units to the right.
- Left Shift: $f(x + h)$ shifts the function $f(x)$ by $h$ units to the left.
Counterintuitive Direction Rule:
- Subtracting a positive value inside the function argument ( $x - h$ ) moves the graph to the right (positive $x$ -direction).
- Adding a positive value inside the function argument ( $x + h$ ) moves the graph to the left (negative $x$ -direction).
Evaluation of Horizontal Shifts:
- Consider an original function $f(x)$ evaluated at discrete points:
- At $x = 0$ , $f(0) = 1$
- At $x = 1$ , $f(1) = 2$
- Evaluating $f(x - 1) = x - 1$ across consecutive inputs:
- At $x = 0$ : $0 - 1 = -1$
- At $x = 1$ : $1 - 1 = 0$
- At $x = 2$ : $2 - 1 = 1$
- At $x = 3$ : $3 - 1 = 2$
- Output values repeat their exact sequence, but are delayed across the domain to higher $x$ -values.

Impact on Domain and Range:
- Horizontal shifts modify the inputs required to produce outputs, altering the domain by $h$ units.
- The set of output values produced remains identical, so the range remains completely unchanged.
Square Root Function Case Study:
- Base function: $f(x) = \sqrt{x}$
- Base Domain: $[0, \infty)$ (since real square roots require non-negative inputs $x \ge 0$ ).
- Base Range: $[0, \infty)$
- Shifted function by $2$ units to the right: $g(x) = \sqrt{x - 2}$
- To maintain valid real outputs, inputs must satisfy $x - 2 \ge 0 \implies x \ge 2$
- New Domain: $[2, \infty)$ (domain shifted right by $h = 2$ units).
- New Range: $[0, \infty)$ (range remains unchanged).
Summary of Functional Shifts:
- Vertical Shift ( $f(x) \pm k$ ): Domain is unchanged; Range shifts by $k$ units.
- Horizontal Shift ( $f(x \pm h)$ ): Domain shifts by $h$ units; Range is unchanged.

Given Discrete Data Points for $f(x)$ :
- $f(-2) = -1$
- $f(-1) = 0$
- $f(0) = 3$
- $f(1) = 2$
- $f(3) = -2$
Constructing the Shifted Function $g(x) = f(x - 1)$ :
- The function $g(x)$ represents a horizontal shift of $f(x)$ by $1$ unit to the right.
- The domain of $g(x)$ starts at $x = -1$ (shifting the original starting domain value $x = -2$ rightward by $1$ unit).
- Evaluation of $g(-1)$ : $g(-1) = f(-1 - 1) = f(-2) = -1$

Determining the Output of $g(0)$ for $g(x) = f(x - 1)$ :
- Prompt: What is the precise output value of $g(0)$ given $g(x) = f(x - 1)$ and the provided tabular values for $f(x)$ ?
- Proposed Student Responses: Options considered included $2$ , $-1$ , $-4$ , $0$ , and $1$
- Derivation and Solution:
- Substitute $x = 0$ directly into the defined relationship $g(x) = f(x - 1)$ : $g(0) = f(0 - 1) = f(-1)$
- Refer to the baseline tabular values for $f(x)$ , where $f(-1) = 0$
- Conclude that $g(0) = 0$