Which Is The Best Description Of The Graph Of The Function $f(x) = 60\left(\frac{1}{3}\right)^x$?A. The Graph Has An Initial Value Of 20, And Each Successive Term Is Determined By Subtracting $\frac{1}{3}$.B. The Graph Has An

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Introduction to the Function

The given function is f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x. This function represents an exponential decay, where the base is 13\frac{1}{3} and the initial value is 6060. The function can be rewritten as f(x)=60exln⁡(13)f(x) = 60e^{x\ln\left(\frac{1}{3}\right)}, where ee is the base of the natural logarithm and ln⁡\ln is the natural logarithm function.

Understanding Exponential Decay

Exponential decay is a type of function where the value decreases rapidly as the input increases. In this case, the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents a decay where the value decreases by a factor of 13\frac{1}{3} for each successive term.

Analyzing the Graph

To understand the graph of the function, let's analyze the behavior of the function as xx increases. As xx increases, the value of (13)x\left(\frac{1}{3}\right)^x decreases rapidly. This means that the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x will also decrease rapidly as xx increases.

Comparing with the Options

Now, let's compare the graph of the function with the options provided.

Option A: The graph has an initial value of 20, and each successive term is determined by subtracting 13\frac{1}{3}

This option is incorrect because the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, where the value decreases by a factor of 13\frac{1}{3} for each successive term, not by subtracting 13\frac{1}{3}.

Option B: The graph has an initial value of 60, and each successive term is determined by multiplying by 13\frac{1}{3}

This option is correct because the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, where the value decreases by a factor of 13\frac{1}{3} for each successive term.

Conclusion

In conclusion, the best description of the graph of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x is that it has an initial value of 60, and each successive term is determined by multiplying by 13\frac{1}{3}. This represents an exponential decay, where the value decreases rapidly as the input increases.

Key Takeaways

  • The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay.
  • The value decreases by a factor of 13\frac{1}{3} for each successive term.
  • The graph has an initial value of 60.
  • The graph decreases rapidly as the input increases.

Real-World Applications

Exponential decay has many real-world applications, such as:

  • Radioactive decay: The decay of radioactive materials follows an exponential decay curve.
  • Population growth: The growth of a population can be modeled using an exponential growth curve, but the decay of a population can be modeled using an exponential decay curve.
  • Chemical reactions: The rate of a chemical reaction can be modeled using an exponential decay curve.

Final Thoughts

In conclusion, the graph of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, where the value decreases by a factor of 13\frac{1}{3} for each successive term. This has many real-world applications, and understanding exponential decay is crucial in many fields of study.

Q: What is the initial value of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x?

A: The initial value of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x is 60. This is the value of the function when x=0x = 0.

Q: How does the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x change as xx increases?

A: The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x decreases rapidly as xx increases. This is because the base of the function is 13\frac{1}{3}, which is less than 1.

Q: What is the rate of decay of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x?

A: The rate of decay of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x is a factor of 13\frac{1}{3} for each successive term. This means that the value of the function decreases by a factor of 13\frac{1}{3} for each unit increase in xx.

Q: How does the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x compare to the function f(x)=60(1−x)f(x) = 60(1 - x)?

A: The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, while the function f(x)=60(1−x)f(x) = 60(1 - x) represents a linear decay. The exponential decay curve is steeper than the linear decay curve, meaning that the value of the function decreases more rapidly as xx increases.

Q: What is the domain of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x?

A: The domain of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x is all real numbers, or (−∞,∞)(-\infty, \infty). This means that the function is defined for all values of xx.

Q: What is the range of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x?

A: The range of the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x is all positive real numbers, or (0,∞)(0, \infty). This means that the value of the function is always positive.

Q: How does the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x relate to the concept of half-life?

A: The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, which is similar to the concept of half-life. In the context of radioactive decay, the half-life is the time it takes for the amount of a substance to decrease by half. The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x can be used to model this type of decay.

Q: Can the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x be used to model population growth?

A: No, the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, which is not suitable for modeling population growth. Population growth is typically modeled using an exponential growth curve, such as f(x)=60(1+x)f(x) = 60(1 + x).

Q: How does the function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x relate to the concept of compound interest?

A: The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x represents an exponential decay, which is similar to the concept of compound interest. In the context of finance, compound interest is the interest earned on both the principal amount and any accrued interest over time. The function f(x)=60(13)xf(x) = 60\left(\frac{1}{3}\right)^x can be used to model this type of interest.