Problem 33 - Algebra for College Students

Question

Are the graphs of \(f(x)=2 \sqrt{x}\) and \(g(x)=\sqrt{2 x}\) identical? Defend your answer.

Step-by-Step Solution

Answer

The graphs are not identical since \( f(x) = 2 \sqrt{x} \neq \sqrt{2} \times \sqrt{x} = g(x) \).

Step 1: Understand the Functions

Let's consider the two given functions. The function \( f(x) = 2 \sqrt{x} \) is a multiple of a square root, and the function \( g(x) = \sqrt{2x} \) is the square root of a product. We need to analyze their structures to determine if their graphs are identical.

Step 2: Analyze Domain of Functions

Both functions \( f(x) \) and \( g(x) \) involve square roots, which require non-negative values. Therefore, their domain is \( x \geq 0 \). Both functions exist for the same range of \( x \), hence have the same domain.

Step 3: Simplify Function Expressions

Simplify and compare the outputs of functions: - For \( f(x) = 2 \sqrt{x} \), there is no further simplification;- For \( g(x) = \sqrt{2x} \), use properties of square roots: \( g(x) = \sqrt{2}\sqrt{x} \) results in \( g(x) = \sqrt{2}\times\sqrt{x} \).

Step 4: Compare Function Values

Compare \( f(x) = 2 \sqrt{x} \) and \( g(x) = \sqrt{2} \times \sqrt{x} \). Note that \( 2 \) and \( \sqrt{2} \) are different,- Since \( 2 eq \sqrt{2} \), we see that at any given \( x \), \( f(x) eq g(x) \).

Step 5: Interpret Results in Terms of Graphs

Because \( 2 eq \sqrt{2} \), the two functions yield different values for any \( x \geq 0 \), leading to different graph profiles. The coefficient in front of the square root in each function determines their respective rates of growth.

Key Concepts

Function Comparison

When comparing two algebraic functions, it's important to focus on their structures. Here, we are looking at \( f(x) = 2 \sqrt{x} \) and \( g(x) = \sqrt{2x} \). At a glance, these functions might seem similar because they both involve the square root operation. However, upon closer inspection, their differences become apparent.

• Structure of Function: The function \( f(x) \) directly multiplies \( \sqrt{x} \) by 2. Conversely, \( g(x) \) takes the square root of \( 2x \), which involves multiplying the variable inside the root.
  
• Simplification: Simplifying \( g(x) \) using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \), we get \( \sqrt{2} \cdot \sqrt{x} \), showing explicitly that these functions have different scaling factors: 2 vs \( \sqrt{2} \).
  
• Function Values: Because 2 is not equal to \( \sqrt{2} \), the functions produce different outputs for any given \( x \).
  

Function comparison is vital to understand because it shows how small differences in expressions can lead to different behaviors.

Graph Analysis

Graph analysis involves understanding how different functions are visually represented in a coordinate space. For \( f(x) = 2 \sqrt{x} \) and \( g(x) = \sqrt{2x} \), the analysis of their graphs reveals key differences influenced by their coefficients and expressions.

• Growth Rate: The coefficient in front of the square root plays a major role in growth. \( f(x) \) grows at twice the rate of \( \sqrt{x} \), while \( g(x) \) grows at a rate of \( \sqrt{2} \) times \( \sqrt{x} \). This results in distinct slopes and shapes for their graphs.
  
• Shape and Size: Despite starting from the same point at \( x = 0 \), the rates at which they increase make their graphs diverge. \( f(x) \) will generally rise more steeply compared to \( g(x) \).
  
• Identical Graphs? Since they have different rates of increase, the two functions do not form identical graphs. Each graph represents a different path due to their unique expressions.
  

This analysis underlines the importance of understanding how coefficients and expressions within functions affect their overarching graphical representation.

Domain and Range

The domain and range are critical components that define where a function is applicable and what values it can output. Both \( f(x) = 2 \sqrt{x} \) and \( g(x) = \sqrt{2x} \) have a similar domain but differ in their output range because of their functional differences.

• Domain: Since both functions involve a square root, their domain is restricted to non-negative values of \( x \) (i.e., \( x \geq 0 \)). This is because we cannot square root negative numbers within real number calculations.
  
• Range: Even though the domain is the same, their output ranges differ due to distinct coefficients. \( f(x) = 2 \sqrt{x} \) allows values beginning from 0 upwards, but increasing more steeply compared to \( g(x) \). In contrast, \( g(x) = \sqrt{2x} \) also outputs from 0 upwards but does so at a relatively slower rate.
  

Understanding both the domain and range helps in picturing how far a function stretches in any direction and the limits of its applicability and output.