## Introduction

If you’ve been struggling with Lesson 14 Homework 4.3, worry no more! In this article, we will reveal the answer key for this assignment, allowing you to ace your work and gain a better understanding of the concepts covered. So, let’s dive in and discover the solutions together!

## Lesson 14 Homework 4.3: Answer Key Revealed

### Question 1:

The first question of Lesson 14 Homework 4.3 asked you to solve a mathematical equation involving logarithms. The equation was as follows:

[2log_3(x+1) – log_3(x-2) = 1]To solve this equation, we need to use some logarithmic properties. Start by combining the logarithms on the left-hand side using the rule of logarithmic subtraction. This gives us:

[log_3left(frac{{(x+1)^2}}{{(x-2)}}right) = 1]Next, we can rewrite the equation using exponential form:

[frac{{(x+1)^2}}{{(x-2)}} = 3^1]Simplifying the right-hand side gives us:

[frac{{(x+1)^2}}{{(x-2)}} = 3]Expanding the numerator on the left-hand side gives us:

[frac{{x^2 + 2x + 1}}{{(x-2)}} = 3]Cross-multiplying and simplifying gives us the quadratic equation:

[x^2 + 2x + 1 = 3(x-2)]Solving this quadratic equation, we find that the solutions are (x = -2) and (x = 4).

### Question 2:

The second question of Lesson 14 Homework 4.3 involved finding the domain and range of a given function. Let’s take a look at the function and how to determine its domain and range.

The function is given as:

[f(x) = sqrt{3-x}]To find the domain, we need to consider the values that (x) can take. In this case, the expression under the square root must be greater than or equal to zero since we cannot take the square root of a negative number. Solving the inequality (3-x geq 0), we find that the domain of the function is (x leq 3).

To find the range, we need to consider the values that the function can output. Since (sqrt{3-x}) will always be a non-negative value, the range of the function is all real numbers greater than or equal to zero.

With the answer key for Lesson 14 Homework 4.3 revealed, you can now confidently tackle these assignments. Remember to always show your work and explain your steps, as it will help you understand the concepts better. Don’t forget to practice and ask for help when needed. Good luck with your future assignments!