Problem 23 - College Algebra with Modeling and Visualization

Question

Comparing Growth Which function becomes larger for \(0 \leq x \leq 10: f(x)=2^{x}\) or \(g(x)=x^{2} ?\)

Step-by-Step Solution

Answer

For \( 0 \leq x \leq 10 \), \( f(x) = 2^x \) becomes larger than \( g(x) = x^2 \) starting from \( x = 5. \)

Step 1: Understanding the Functions

We need to compare the growth of two functions: \( f(x) = 2^x \) and \( g(x) = x^2 \) for values of \( x \) ranging from 0 to 10. \( f(x) = 2^x \) is an exponential function while \( g(x) = x^2 \) is a quadratic function.

Step 2: Evaluate Both Functions at Critical Points

Evaluate both functions at critical points, such as \( x = 0, x = 1, x = 2, \) and continue towards \( x = 10 \):- At \( x = 0 \), \( f(x) = 2^0 = 1 \) and \( g(x) = 0^2 = 0 \).- At \( x = 1 \), \( f(x) = 2^1 = 2 \) and \( g(x) = 1^2 = 1 \).- At \( x = 2 \), \( f(x) = 2^2 = 4 \) and \( g(x) = 2^2 = 4 \).

Step 3: Compare Both Functions for Increasing x

Continue evaluating for higher \( x \):- At \( x = 3 \), \( f(x) = 2^3 = 8 \) while \( g(x) = 3^2 = 9 \). Here, \( g(x) > f(x) \).- At \( x = 4 \), \( f(x) = 2^4 = 16 \) and \( g(x) = 4^2 = 16 \). Both are equal.- At \( x = 5 \), \( f(x) = 2^5 = 32 \) and \( g(x) = 5^2 = 25 \). Now, \( f(x) > g(x) \).- Continue similarly for \( x = 6, 7, 8, 9, 10 \), noting that \( f(x) \) continues to grow faster.

Step 4: Conclusion on Growth within Range

Around \( x = 5 \), the exponential function \( f(x) = 2^x \) overtakes the quadratic function \( g(x) = x^2 \) and grows faster for all points \( x > 4 \) up to 10.

Key Concepts

Quadratic Functions

Quadratic functions are an essential concept in mathematics and appear in various real-world situations. The general form of a quadratic function is \( g(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).The quadratic function \( g(x) = x^2 \) given in the exercise is a simple form, with \( a = 1 \) and \( b = c = 0 \).

• The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \). In this case, it opens upwards.
• The vertex of a parabola \( f(x) = x^2 \) is at the origin \((0, 0)\).
• Quadratic functions are symmetrical about a vertical line through their vertex.

Quadratic functions have several interesting properties. They reach their minimum or maximum at the vertex, and this value plays a critical role when comparing with other functions, like in the exercise given.

Function Comparison

Comparing functions involves analyzing their behavior over a specific range of values.In the provided exercise, we compare the quadratic function \( g(x) = x^2 \) with the exponential function \( f(x) = 2^x \).

• Initially, at \( x = 0 \), the exponential function has a higher value, \( f(x) = 1 \) while \( g(x) = 0 \).
• At \( x = 2 \), both functions equate, \( f(x) = g(x) = 4 \).
• For \( x = 3 \) and \( x = 4 \), the quadratic function takes the lead.

The exercise demonstrates that around \( x = 5 \), the exponential function starts growing faster than the quadratic function.In function comparison, observing points of intersection and growth rates is key.This allows us to predict which function will dominate over certain intervals.

Function Growth

Understanding the growth of functions helps us predict their long-term behavior. Quadratic functions, like \( g(x) = x^2 \), grow at a polynomial rate, characterized by how their outputs become larger as \( x \) increases. Exponential functions, such as \( f(x) = 2^x \), grow very quickly as \( x \) becomes larger. They start slower than quadratic functions initially but eventually overshadow them due to their rapid rate of increase:

• Exponential growth implies multiplicative changes. Each increase by 1 in \( x \) results in the function's value doubling.
• While quadratic functions add a linear sequence of growth (like \( x, 2x, 3x, ... \)), exponential growth stacks that on top, creating a much steeper curve.

This concept of growth becomes prominent in the exercise, where the exponential function \( f(x) \) surpasses \( g(x) \) around \( x = 5 \), demonstrating exponential growth's overpowering nature.Understanding how these functions grow is crucial in many fields, from population modeling to finance, where such patterns frequently emerge.