Problem 14 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=\sqrt{x-1}+3 $$

Step-by-Step Solution

Answer

Shift \( \sqrt{x} \) right by 1, and up by 3; graph starts at \( (1, 3) \).

Step 1: Identify the Base Function

The base function is the square root function \( g(x) = \sqrt{x} \). It is defined only for \( x \geq 0 \) and has a graph that starts at the origin and extends to the right on the x-axis as a curve.

Step 2: Determine the Domain of the Function

Since the function \( f(x) = \sqrt{x-1} + 3 \) involves the expression \( \sqrt{x-1} \), it is only defined for \( x - 1 \geq 0 \). Solve \( x - 1 \geq 0 \) to get \( x \geq 1 \). Therefore, the domain of \( f(x) \) is \( x \geq 1 \).

Step 3: Apply the Horizontal Shift

The function \( g(x) = \sqrt{x} \) is shifted to the right by 1 unit, resulting in \( h(x) = \sqrt{x-1} \). This horizontal transformation offsets the entire graph of the square root function to start from \( x=1 \) instead of \( x=0 \).

Step 4: Apply the Vertical Shift

After the horizontal shift, the function is \( h(x) = \sqrt{x-1} \). Then, apply the vertical shift of 3 units upwards to get the complete function \( f(x) = \sqrt{x-1} + 3 \). This moves each point on the graph of \( h(x) \) up by 3 units.

Step 5: Sketch the Graph

Start the graph at the point \( (1, 3) \) because this is where the function is defined. Plot a point there, and then draw the curve by following the standard shape of the square root function, starting at \( (1, 3) \) and extending to the right, tracking upwards.

Key Concepts

Domain and Range

Understanding the domain and range of a function is essential for sketching its graph. The **domain** refers to all the possible input values ("x" values) for which the function is defined.
In the context of the function \( f(x)=\sqrt{x-1}+3 \), the domain is determined by the expression under the square root, \( x-1 \), which must be non-negative. Solving \( x-1 \geq 0 \) gives \( x \geq 1 \). This means the domain of \( f(x) \) is all real numbers \( x \) such that \( x \geq 1 \).

On the other hand, the **range** of a function is the set of all possible output values ("y" values). For the base square root function \( g(x) = \sqrt{x} \), the range is \( y \geq 0 \). However, the function \( f(x)=\sqrt{x-1}+3 \) is shifted up by 3 units.
Therefore, the smallest value \( y \) can be is \( 3 \), making the range \( y \geq 3 \). In conclusion, while sketching the graph, it's important to account for these possible values that the function can accept and produce.

Function Transformations

Transforming functions involves altering their graphs through shifts, stretches, compressions, or reflections. **Horizontal shifts** occur when we add or subtract inside the function's rule. In \( f(x) = \sqrt{x-1} + 3 \), the \( x-1 \) indicates a horizontal shift to the right by 1 unit from the graph of the basic square root function \( g(x) = \sqrt{x} \).
This moves the starting point from the origin \( (0, 0) \) to \( (1, 0) \).

A **vertical shift** is introduced by adding or subtracting values outside the function rule. The "+3" in \( \sqrt{x-1}+3 \) shifts the entire graph up by 3 units.

• This results in a new starting point at \( (1, 3) \).

Understanding these shifts helps you modify the graph easily and predict accurately how the graph will appear without needing to plot multiple points.

Square Root Function

The square root function, one of the fundamental mathematical functions, is typically defined as \( g(x) = \sqrt{x} \).
It is characterized by its distinct curve, which starts at the origin \( (0,0) \) and gently rises to the right.

This occurs because the square root of non-negative numbers gradually increases, reflecting a slower rate of growth in the function compared to linear or quadratic functions.

In our transformed function \( f(x) = \sqrt{x-1} + 3 \), these changes are influenced by the horizontal and vertical shifts explained earlier.

• This function does not exist before \( x=1 \), due to the horizontal shift.
• It starts at the point \( (1, 3) \) following the upward shift.

Knowing the basic pattern and behavior of \( g(x) = \sqrt{x} \) ensures a solid understanding of how transformations affect the function, allowing you to sketch it accurately and with confidence.