If $f(x) = 1.5^x + 1$, What Is The Value Of $f(2)$?

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Introduction

In mathematics, exponential equations are a fundamental concept that deals with the relationship between a variable and its exponent. These equations are used to model real-world situations, such as population growth, financial investments, and chemical reactions. In this article, we will focus on solving exponential equations, specifically the equation $f(x) = 1.5^x + 1$, and find the value of $f(2)$.

Understanding Exponential Functions

An exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The base $a$ determines the rate of growth or decay of the function. In the given equation $f(x) = 1.5^x + 1$, the base is $1.5$, which means that the function will grow exponentially as $x$ increases.

Evaluating the Function at x = 2

To find the value of $f(2)$, we need to substitute $x = 2$ into the equation $f(x) = 1.5^x + 1$. This means that we will replace $x$ with $2$ and calculate the result.

import math

def f(x):
return 1.5**x + 1


result = f(2)
print(result)

Step-by-Step Solution

To solve this problem, we will follow these steps:

  1. Substitute x = 2 into the equation: Replace $x$ with $2$ in the equation $f(x) = 1.5^x + 1$.
  2. Calculate the value of 1.5^2: Use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression.
  3. Add 1 to the result: Finally, add $1$ to the result to get the final answer.

Step 1: Substitute x = 2 into the equation

f(2)=1.52+1f(2) = 1.5^2 + 1

Step 2: Calculate the value of 1.5^2

Using the property of exponents, we can simplify the expression as follows:

1.52=1.5⋅1.5=2.251.5^2 = 1.5 \cdot 1.5 = 2.25

Step 3: Add 1 to the result

Finally, we add $1$ to the result to get the final answer:

f(2)=2.25+1=3.25f(2) = 2.25 + 1 = 3.25

Conclusion

In this article, we solved the exponential equation $f(x) = 1.5^x + 1$ and found the value of $f(2)$. We used the property of exponents to simplify the expression and calculated the result step by step. The final answer is $f(2) = 3.25$.

Real-World Applications

Exponential equations have many real-world applications, such as:

  • Population growth: Exponential equations can be used to model population growth, where the population grows at a constant rate.
  • Financial investments: Exponential equations can be used to calculate the future value of an investment, where the interest rate is constant.
  • Chemical reactions: Exponential equations can be used to model chemical reactions, where the rate of reaction is constant.

Common Mistakes

When solving exponential equations, it's common to make mistakes such as:

  • Forgetting to simplify the expression: Make sure to simplify the expression using the properties of exponents.
  • Not adding 1 to the result: Don't forget to add $1$ to the result to get the final answer.

Tips and Tricks

Here are some tips and tricks to help you solve exponential equations:

  • Use the property of exponents: Use the property of exponents to simplify the expression.
  • Calculate the value of the base: Calculate the value of the base before substituting it into the equation.
  • Add 1 to the result: Don't forget to add $1$ to the result to get the final answer.

Conclusion

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves an exponential function, which is a function that has the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to follow these steps:

  1. Substitute the given value into the equation: Replace the variable with the given value.
  2. Simplify the expression: Use the properties of exponents to simplify the expression.
  3. Calculate the result: Calculate the result of the simplified expression.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation is a mathematical equation that involves an exponential function, while a linear equation is a mathematical equation that involves a linear function. Exponential equations have a base that is raised to a power, while linear equations have a slope and a y-intercept.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, make sure to check your calculator's settings to ensure that it is set to the correct mode (e.g. scientific mode).

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the following properties of exponents:

  • Product of powers: $a^m \cdot a^n = a^{m+n}$
  • Power of a power: $(am)n = a^{m \cdot n}$
  • Quotient of powers: $\frac{am}{an} = a^{m-n}$

Q: What is the value of $f(3)$ if $f(x) = 2^x + 1$?

A: To find the value of $f(3)$, we need to substitute $x = 3$ into the equation $f(x) = 2^x + 1$.

f(3)=23+1=8+1=9f(3) = 2^3 + 1 = 8 + 1 = 9

Q: Can I use a graphing calculator to visualize an exponential function?

A: Yes, you can use a graphing calculator to visualize an exponential function. Graphing calculators can help you visualize the behavior of an exponential function and identify key features such as the x-intercept, y-intercept, and asymptotes.

Q: How do I determine the domain and range of an exponential function?

A: To determine the domain and range of an exponential function, you need to consider the following:

  • Domain: The domain of an exponential function is all real numbers, unless the function has a restriction on the domain (e.g. a vertical asymptote).
  • Range: The range of an exponential function is all positive real numbers, unless the function has a restriction on the range (e.g. a horizontal asymptote).

Q: Can I use an exponential function to model real-world phenomena?

A: Yes, you can use an exponential function to model real-world phenomena such as population growth, financial investments, and chemical reactions.

Q: How do I choose the correct base for an exponential function?

A: To choose the correct base for an exponential function, you need to consider the following:

  • Rate of growth: Choose a base that represents the rate of growth or decay of the function.
  • Initial value: Choose a base that represents the initial value of the function.

Q: Can I use an exponential function to model a situation with a negative rate of growth?

A: Yes, you can use an exponential function to model a situation with a negative rate of growth. However, you need to choose a base that represents the negative rate of growth.

Conclusion

In conclusion, exponential equations are a fundamental concept in mathematics that deals with the relationship between a variable and its exponent. In this article, we answered frequently asked questions about exponential equations, including how to solve them, simplify them, and visualize them. We also discussed the importance of choosing the correct base for an exponential function and how to model real-world phenomena using exponential functions.