Problem 5 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ h=\\{(x, y) \mid y=\sqrt{3 x-4}\\} $$

Step-by-Step Solution

Answer

Domain: \( x \in \left[ \frac{4}{3}, \infty \right) \), Range: \( y \in [0, \infty) \).

Step 1 - Determine the Domain

The domain consists of the values of x for which the function is defined. For the function \( y = \sqrt{3x-4} \), the expression under the square root must be non-negative: \[ 3x - 4 \geq 0 \] Solving for x: \[ 3x \geq 4 \] \[ x \geq \frac{4}{3} \] Therefore, the domain of the function is \( x \in \left[ \frac{4}{3}, \infty \right) \).

Step 2 - Determine the Range

The range consists of the possible values of y. Since the square root function returns non-negative values only, and the expression under the square root can be any non-negative number, the range of the function is \[ y \geq 0 \] Thus, the range is \( y \in [0, \infty) \).

Step 3 - Sketch the Graph

To sketch the graph of the function:\( y = \sqrt{3x-4} \): 1. Plot the point where the function begins, which is at \( x = \frac{4}{3} \), \( y = 0 \). 2. For larger values of x, compute corresponding values of y to get additional points: For example, if \( x = 2 \), then: \[ y = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \] Plot this point. 3. Draw a curve starting at \( (\frac{4}{3}, 0) \) and rising steadily to the right.

Key Concepts

Function Domain

The domain of a function refers to all possible input values (x-values) that make the function work without any issues. In this problem, we are given the function defined by:
\( \ y = \sqrt{3x - 4} \)
For this function to have real output values for y, we need to ensure that the values under the square root are non-negative. This is because the square root of a negative number is not a real number. Therefore, we need to solve the inequality:
\( \ 3x - 4 \geq 0 \)
Solving this inequality step-by-step, we get:
\( \ 3x \geq 4 \)
\( x \geq \frac{4}{3} \)
This implies that x must be equal to or greater than \( \ \frac{4}{3} \) . In interval notation, the domain of the function is:
\( \ x \in \left[ \frac{4}{3}, \infty \right) \)
In simpler terms, as long as x is any value starting from \( \ \frac{4}{3} \) or larger, the function is defined.

Function Range

The range of a function is all possible output values (y-values) that the function can produce. Our given function is:
\( \ y = \sqrt{3x - 4} \)
The square root function always yields non-negative results (i.e., zero or positive). Therefore, y must be zero or greater:
\( \ y \geq 0 \)
Since the smallest value under the square root is zero (when x = \( \ \frac{4}{3} \)), y starts from zero and can increase without bound as x increases. Thus, in interval notation, the range of the function is:
\( \ y \in [0, \infty) \)
This tells us that y can take all values starting from 0 and extending to any positive number.

Graphing Functions

Graphing a function helps us visualize the relationship between x and y. For our function
\( \ y = \sqrt{3x - 4} \)
we start by identifying key points. The function begins where x = \( \ \frac{4}{3} \), which gives y = 0. This is our starting point on the graph. To graph this function, follow these steps:

• Plot the point where the function starts: at \( \ (\frac{4}{3}, 0) \).
• Choose other values of x greater than \( \ \frac{4}{3} \) to find corresponding y values. For instance, if x = 2:
  \( y = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \). Plot this point (2, \( \sqrt{2} \)).
• Continue selecting increasing x values to find more points. For example, if x = 3:
  \( y = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \). Plot this point (3, \( \sqrt{5} \)).
• Draw a smooth curve through these plotted points, starting from \( \ (\frac{4}{3}, 0) \) and moving to the right.

As you continue to plot points, you will see that the graph rises steadily and never turns downward. This curve represents the function visually, showing how y increases as x increases beyond \( \ \frac{4}{3} \).