Problem 48 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a linear or nonlinear function. If \(f\) is a linear function, determine if \(f\) is a constant function. Support your answer by graphing \(f\). $$ f(x)=\sqrt{x-1} $$

Step-by-Step Solution

Answer

The function is nonlinear and not constant.

Step 1: Determine the Form of the Function

The function is given as \(f(x) = \sqrt{x-1}\). A linear function is typically in the form \(f(x) = mx + b\) where \(m\) and \(b\) are constants. This function contains a square root, which indicates it is not a simple equation of the form \(mx + b\).

Step 2: Analyze for Constant Function

A constant function has the form \(f(x) = c\) where \(c\) is a constant value for any input \(x\). In \(f(x) = \sqrt{x-1}\), the value of \(f(x)\) changes as \(x\) changes, so it is not a constant function either.

Step 3: Graph the Function

Graphing the function \(f(x) = \sqrt{x-1}\) can provide a visual representation. The square root function has a domain from \(x = 1\) to infinity and an increasing curve starting at \((1,0)\). This confirms that the output changes with the input \(x\), which further supports that it is nonlinear.

Key Concepts

Graphing Functions

When you're graphing functions, you often get a clear visual insight into the behavior of the function. For the function given as \( f(x) = \sqrt{x-1} \), its graph offers the first clue about whether it is linear or nonlinear. To start, the basic shape and behavior of the graph provide valuable information.

• Functions that form a straight line on a plane are linear functions.
• Curved behaviors normally indicate a nonlinear function.

If you draw the graph of \(f(x) = \sqrt{x-1}\), you'll see that it doesn't form a straight line but rather a curve that starts at the point \((1, 0)\) and continues to rise as \(x\) increases. It confirms that the function is nonlinear. This visual representation can be a powerful tool to understand functions at a glance and is often used in solving and checking math exercises.

Function Analysis

Function analysis involves understanding the behavior of a function beyond just its equation. By examining \(f(x) = \sqrt{x-1}\), we seek to identify characteristics such as linearity or constancy.

Linearity Check

First, to determine linearity, consider the form of the equation. Linear functions have the form \(f(x) = mx + b\). In function analysis, identifying terms like squares or square roots immediately indicate nonlinearity. Since \(\sqrt{x-1}\) doesn't follow the pattern \(mx + b\), it's clear this function lacks linearity.

Behavior Over Domain

Analyzing behavior over its domain is also crucial. For \(f(x) = \sqrt{x-1}\), the domain is \(x \geq 1\). Each valid \(x\) value results in a unique \(f(x)\) value, often changing as \(x\) varies. This change in output further confirms that \(f(x)\) is nonlinear. Function analysis lets you thoroughly understand the changes and nature of functions.

Constant Function

A constant function has a unique characteristic: it returns the same value for any input within its domain. In mathematical terms, a constant function has the form \(f(x) = c\). This means the graph of a constant function is a horizontal line across the same \(y\)-value.For \(f(x) = \sqrt{x-1}\), notice how the value of \(f(x)\) changes depending on \(x\). If you plug different \(x\) values into \(\sqrt{x-1}\), you'll observe different outputs. This variability is a clear indication that \(f(x)\) is not a constant function.Understanding constant functions can help recognize functions that lack variation, unlike \(f(x) = \sqrt{x-1}\) which is evidently not constant.