Sure, let's break down the function [tex]\( g(x) = \sqrt{2x + 3} \)[/tex] into a detailed step-by-step explanation.
### Step-by-Step Solution:
1. Identify the Function: The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = \sqrt{2x + 3} \)[/tex].
2. Understand the Components:
- [tex]\( x \)[/tex] is the variable input to the function.
- Inside the square root, the expression [tex]\( 2x + 3 \)[/tex] needs to be evaluated first.
3. Calculate [tex]\( 2x + 3 \)[/tex]:
- For a given value of [tex]\( x \)[/tex], multiply it by 2.
- Add 3 to the result of the multiplication.
4. Square Root Calculation:
- Take the square root of the value obtained from the previous step.
### Example Calculation:
Let's go through an example to clarify these steps.
#### Example 1: Evaluate [tex]\( g(2) \)[/tex].
1. Start with the input value [tex]\( x = 2 \)[/tex].
2. Calculate the expression inside the square root:
[tex]\[
2x + 3 = 2 \cdot 2 + 3 = 4 + 3 = 7
\][/tex]
3. Take the square root of the result:
[tex]\[
g(2) = \sqrt{7}
\][/tex]
#### Example 2: Evaluate [tex]\( g(5) \)[/tex].
1. Start with the input value [tex]\( x = 5 \)[/tex].
2. Calculate the expression inside the square root:
[tex]\[
2x + 3 = 2 \cdot 5 + 3 = 10 + 3 = 13
\][/tex]
3. Take the square root of the result:
[tex]\[
g(5) = \sqrt{13}
\][/tex]
### Generalization:
For any input [tex]\( x \)[/tex]:
1. Compute [tex]\( 2x + 3 \)[/tex].
2. Take the square root of the resulting value.
This process will give you the value of the function [tex]\( g(x) \)[/tex] for any [tex]\( x \)[/tex] you provide.
Remember, since we are dealing with a square root, the expression [tex]\( 2x + 3 \)[/tex] must be non-negative for real-valued results. This means [tex]\( 2x + 3 \geq 0 \implies x \geq -1.5 \)[/tex]. Thus, the domain of [tex]\( g(x) \)[/tex] is [tex]\( x \geq -1.5 \)[/tex].
I hope this clarifies the step-by-step procedure to evaluate the function [tex]\( g(x) = \sqrt{2x + 3} \)[/tex]. If you have any further questions or need additional examples, feel free to ask!
