Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$f(x-3)+1$$

Step-by-Step Solution

Answer

Domain: [2, 5]; Range: [1, 4].

Step 1: Determine the New Domain

To find the domain of the transformed function \(f(x-3)+1\), start with the original domain of \([-1, 2]\). Since \(x-3\) shifts the graph 3 units to the right, we need to set \(x-3 = a\) where \(a\) is in \([-1, 2]\). This implies \(x = a + 3\). Thus, the new domain is obtained by adding 3 to each element of the original domain: \[x \,\text{ranges from} \, -1 + 3 \,\text{to}\, 2 + 3\]making it \([2, 5]\).

Step 2: Determine the New Range

For the range, start with the original range of \([0, 3]\). The transformation \(+1\) shifts all function values up by 1 unit. Thus, add 1 to each element of the original range: \[y \,\text{ranges from} \, 0 + 1 \,\text{to}\, 3 + 1\]making the new range \([1, 4]\).

Step 3: Conclude the Domain and Range

After calculating the transformations: - The new domain of \(f(x-3)+1\) is \([2, 5]\).- The new range of \(f(x-3)+1\) is \([1, 4]\).

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial for analyzing its behavior. The **domain** refers to all the possible input values (usually represented as "x") of a function. In our original function, the domain is after this transformation becomes

• - Finding New Domain: Since the function transforms with "\(- 3\)", each input is effectively moved right by 3. Add 3 to the starting and ending values of the original domain.

This changes the domain from To (2, 5].The **range**, meanwhile, represents all possible output values (usually represented as "y") of a function. For our original function, the range is becomes

• - Finding New Range: The "\(+1\)" in the function shifts every output ymax and ymin upwards by 1, tweaking the original range intervals.

This changes the range from. Undergoing these transformations, the range shifts to Through understanding and transforming these elements, one gets a clear picture of how function transformations affect a graph's position vertically and horizontally. Remember, simple transformations can sometimes be enough to alter all the fundamental characteristics of functions and charts.

Shifts in Graphs

Graph shifts form a fundamental part of function transformations. These shifts can occur either horizontally or vertically, depending on how the equation of a function is structured.

• **Horizontal Shifts**: If you have a function \(f(x)\) and replace x with \((x-c)\), it shifts the graph horizontally.

In our problem, replacing x with \((x-3)\) means the graph shifts "right" by 3 units. Visualize it like moving a slide to the right without altering its shape or size.

• **Vertical Shifts**: When you see a \(+c\) or \(-c\) outside the function, it shifts "up" or "down" accordingly.

In our solution, the \(+1\) indicates an upward shift by 1 unit, elevating all points on the graph higher. These shifts help reframe the perspective from which one views the graph, ensuring better comprehension and correct adjustments to predictions or interpretations of data.

Piecewise Functions

Piecewise functions represent scenarios where a function changes its behavior at certain points within its domain. They are often written like this:\[ f(x) = \begin{cases} \text{expression 1,} & \text{for x in range 1} \ \text{expression 2,} & \text{for x in range 2} \end{cases} \] Piecewise functions show different rules or expressions applied to different intervals of the domain. They are particularly insightful for modeling situations with distinct phases.In the context of transformations like \(f(x-3)+1\), you can think about applying transformations to each piece or interval separately. For example:

• **Handling Domain**: Shifts in the expression could influence which part of the piecewise function is relevant when evaluating a particular range of values.
• **Adapting Range**: The vertical shifts affect the resulting outputs of the entire function.

These transformed segments provide a clearer view on how any set pieces of the function evolve or adapt as the graph shifts. It's just like examining different plates in a balanced meal, except here we tune into mathematical movements.