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Chapter 1: Problem 70

Find the domain of the function \(f(x)=\sqrt{3^{x-1}+5^{x-1}+7^{x-1}-83}\)

Short Answer

Expert verified
The domain can be derived from solving the inequality and will be the set of \(x\) values that make the argument of the root non-negative. Depending on the complexity of the function, different numerical or graphical methods might need to be used to solve this.

Step by step solution

01

Define the inequality

As the function includes a square root, it is necessary to ensure the values inside the root are non-negative. Therefore, set up the inequality \(3^{x-1}+5^{x-1}+7^{x-1} - 83 \geq 0.\)
02

Solve the inequality

This inequality can't be solved algebraically for a general \(x\). Here, a graphical approach or numerical methods would be most accurate. One could set up a few initial guesses and use, for instance, a Newton-Raphson method to find the bounds for \(x\).
03

Identify the domain

After solving the inequality in Step 2, any value of \(x\) that satisfies the inequality will be a part of the domain. The domain will be the set of these \(x\) values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality
To find the domain of a function involving a square root, you need to work with inequalities. This is because a square root is only defined when the expression inside is zero or positive. If the value inside the square root becomes negative, the function does not have real values, which means it is undefined in the real numbers.

Let's take the function: \[f(x) = \sqrt{3^{x-1} + 5^{x-1} + 7^{x-1} - 83}\]To ensure this function is defined, you must ensure: \[3^{x-1} + 5^{x-1} + 7^{x-1} - 83 \geq 0.\]

This inequality ensures the expression inside the square root is non-negative. Setting up and solving this inequality forms the basis of finding the domain for such functions.

  • Real solutions: The inequality yields real solutions where the function exists.
  • Non-negative constraint: Ensures the square root is real and not imaginary.
Square root function
Square root functions involve expressions of the form \(\sqrt{expression}\). For a square root function to be real and defined, the expression inside the square root must be greater than or equal to zero.

Consider the function \[f(x) = \sqrt{3^{x-1} + 5^{x-1} + 7^{x-1} - 83}\]For this function, the expression inside the square root, \[3^{x-1} + 5^{x-1} + 7^{x-1} - 83\],must be zero or positive. If not, the result would be an imaginary number, which typically is not desired when considering real-valued functions.

Steps to handle square root equations effectively:
  • Set the inequality: Ensure the expression inside the root is non-negative.
  • Find critical points: Identify where the expression equals zero.
  • Determine solution range: Establish where the expression maintains non-negativity.

Square root functions are fundamental when working with domains as they set strict conditions for the allowable inputs.
Exponential functions
Exponential functions have the general form \(a^x\), where \(a\) is a positive constant. Exponentials grow rapidly and are essential when dealing with complex inequalities such as the one in this exercise.

In our function, the terms \[3^{x-1}, 5^{x-1},\text{ and } 7^{x-1}\]are exponential expressions. They vary depending on the value of \(x\). Exponential functions have distinct characteristics:

  • Base behavior: For bases more than 1 (e.g., 3, 5, or 7), as \(x\) increases, the function also increases rapidly.
  • Positive values: Regardless of the exponent, exponential functions maintain positive values as long as the base is positive.
  • Domain: The domain of basic exponential functions is all real numbers, meaning they are defined everywhere on the real line.

When finding a domain involving these functions in combination (like here), understanding their graphical behavior assists in setting up or checking solution bounds objectively.

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Most popular questions from this chapter

Find the domain of the function \(f(x)=\sqrt{\log _{10}\left(\log _{10} x\right)-\log _{10}\left(4-\log _{10} 3\right)-\log _{10} 3}\)

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