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Magazine Chapter 4: Problem 44
Compare the graphs of the power function \(f\) and exponential function \(g\) by evaluating both of them for \(x=0,1,2,3,4,6,8,\) and 10 Then draw the graphs of \(f\) and \(g\) on the same set of axes. \(f(x)=x^{4} ; \quad g(x)=4^{x}\)
Short Answer
Expert verified
The exponential function \( g(x) = 4^x \) grows much faster than the power function \( f(x) = x^4 \) for large \( x \).
Step by step solution
01
Evaluate Power Function
Calculate the values of the power function \( f(x) = x^4 \) for each given value of \( x \):- \( f(0) = 0^4 = 0 \)- \( f(1) = 1^4 = 1 \)- \( f(2) = 2^4 = 16 \)- \( f(3) = 3^4 = 81 \)- \( f(4) = 4^4 = 256 \)- \( f(6) = 6^4 = 1296 \)- \( f(8) = 8^4 = 4096 \)- \( f(10) = 10^4 = 10000 \)
02
Evaluate Exponential Function
Calculate the values of the exponential function \( g(x) = 4^x \) for each given value of \( x \):- \( g(0) = 4^0 = 1 \)- \( g(1) = 4^1 = 4 \)- \( g(2) = 4^2 = 16 \)- \( g(3) = 4^3 = 64 \)- \( g(4) = 4^4 = 256 \)- \( g(6) = 4^6 = 4096 \)- \( g(8) = 4^8 = 65536 \)- \( g(10) = 4^{10} = 1048576 \)
03
Compare Values
Compare the calculated values of \( f(x) \) and \( g(x) \) for each \( x \):- At \( x = 0 \), \( f(x) = 0 \), \( g(x) = 1 \)- At \( x = 1 \), \( f(x) = 1 \), \( g(x) = 4 \)- At \( x = 2 \), \( f(x) = 16 \), \( g(x) = 16 \)- At \( x = 3 \), \( f(x) = 81 \), \( g(x) = 64 \)- At \( x = 4 \), \( f(x) = 256 \), \( g(x) = 256 \)- At \( x = 6 \), \( f(x) = 1296 \), \( g(x) = 4096 \)- At \( x = 8 \), \( f(x) = 4096 \), \( g(x) = 65536 \)- At \( x = 10 \), \( f(x) = 10000 \), \( g(x) = 1048576 \)
04
Observe Graphical Behavior
Plot the values of both functions on a graph:- Power function \( f(x) = x^4 \) grows polynomially and starts slower but catches up to and surpasses \( g(x) \) at \( x \leq 4 \).- Exponential function \( g(x) = 4^x \) grows much faster for \( x > 4 \), dominating \( f(x) \) as \( x \) increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Function Growth
Power functions are mathematical expressions where a variable, such as \( x \), is raised to a constant power. In this example, our power function is \( f(x) = x^4 \). This specific form refers to it as a polynomial function of degree 4.
The growth of power functions is initially quite slow, especially for lower degree polynomials. They tend to increase more rapidly as \( x \) increases but still maintain a predictable rate.
Key characteristics of power functions include:
The growth of power functions is initially quite slow, especially for lower degree polynomials. They tend to increase more rapidly as \( x \) increases but still maintain a predictable rate.
Key characteristics of power functions include:
- The base of the power (\( x \) here) starting at zero remains zero.
- For positive values, the output increases with increasing inputs.
- The rate of increase becomes noticeable beyond certain points鈥攅specially as \( x \) becomes large.
Exponential Function Growth
Exponential functions are characterized by the variable being in the exponent. In the function \( g(x) = 4^x \), the number 4 is the base, and the growth is driven by raising this base to the power of \( x \).
Unlike power functions, exponential functions experience much faster growth. Initially, they may grow slowly but shortly after, minor changes in \( x \) lead to significant increases in the function鈥檚 value.
Important points to note:
Unlike power functions, exponential functions experience much faster growth. Initially, they may grow slowly but shortly after, minor changes in \( x \) lead to significant increases in the function鈥檚 value.
Important points to note:
- At \(x=0\), any number raised to the power of zero equals 1.
- The value grows very quickly as \(x\) becomes positive.
- Exponential functions tend to dominate power functions as \(x\) increases sufficiently.
Graphing Functions
Graphing power and exponential functions allows you to visualize their growth differently. By plotting \(f(x) = x^4\) and \(g(x) = 4^x\), we can see how their values compare directly.
Steps to graph:
Steps to graph:
- Calculate the function values for selected points (e.g., \(x = 0, 1, 2, 3, 4, 6, 8, 10\)).
- Plot the resulting points on a graph using Cartesian coordinates.
- Draw smooth curves through the points for both functions.
Function Evaluation
Function evaluation involves calculating the value of a function for specific values of \( x \). This practical step is crucial for both understanding and graphing functions' behaviors.
For our power function \( f(x) = x^4 \), and exponential function \( g(x) = 4^x \), we evaluated them at values of \(x = 0, 1, 2, 3, 4, 6, 8, 10\).
Some tips on effective evaluation include:
For our power function \( f(x) = x^4 \), and exponential function \( g(x) = 4^x \), we evaluated them at values of \(x = 0, 1, 2, 3, 4, 6, 8, 10\).
Some tips on effective evaluation include:
- Substitute each value of \(x\) into both functions to find the outputs.
- Use a calculator for larger powers if necessary.
- Tabulate values to clearly see trends and differences.
Comparative Analysis of Functions
A side-by-side comparison of power and exponential functions provides insight into their distinct characteristics and growth behaviors. Let's analyze the functions \( f(x) = x^4 \) and \( g(x) = 4^x \) together.
By comparing values:
By comparing values:
- For \(x = 0\), \(f(x)\) starts at 0 while \(g(x)\) is higher at 1.
- Both functions match at some points (e.g., \(x = 2,4\)).
- The exponential function \(g(x)\) quickly surpasses the power function \(f(x)\) after a certain \(x\).
