Graph the function [tex]g ( x )=\sqrt{7 x -6}[/tex]. Choose the correct graph below.
Answer (1)
Determine the domain of the function: x ≥ 7 6 .
Find the x-intercept: x = 7 6 .
Calculate additional points: ( 1 , 1 ) and ( 2 , 2 2 ) .
Analyze the behavior: The function starts at ( 7 6 , 0 ) and increases to the right. The correct graph is C .
Explanation
Analyzing the Function The problem asks us to identify the correct graph of the function g ( x ) = 7 x − 6 . To do this, we need to determine the domain of the function, find the x-intercept, and analyze the general behavior of the graph.
Determining the Domain First, let's find the domain of the function. Since we have a square root, the expression inside the square root must be non-negative: 7 x − 6 ≥ 0 Solving for x , we get: 7 x ≥ 6 x ≥ 7 6 So, the domain of the function is x ≥ 7 6 .
Finding the X-Intercept Next, let's find the x-intercept. This is the point where g ( x ) = 0 : 7 x − 6 = 0 Squaring both sides, we get: 7 x − 6 = 0 Solving for x , we get: 7 x = 6 x = 7 6 So, the x-intercept is at x = 7 6 , which is approximately 0.857 . At this point, g ( x ) = 0 .
Calculating Additional Points Now, let's find a few additional points to understand the behavior of the function. We already know that when x = 7 6 , g ( x ) = 0 . Let's calculate g ( x ) for x = 1 and x = 2 :
For x = 1 : g ( 1 ) = 7 ( 1 ) − 6 = 1 = 1 So, the point ( 1 , 1 ) is on the graph.
For x = 2 : g ( 2 ) = 7 ( 2 ) − 6 = 14 − 6 = 8 = 2 2 ≈ 2.83 So, the point ( 2 , 2 2 ) is on the graph.
Analyzing the Graph's Behavior The function starts at x = 7 6 with g ( x ) = 0 , and as x increases, g ( x ) also increases. The graph should start at ( 7 6 , 0 ) and increase to the right. Based on this analysis, we can conclude that the correct graph is the one that starts at x = 7 6 and increases.
Final Answer Based on the domain, x-intercept, and the increasing behavior of the function, the correct graph is C .
Examples
Understanding square root functions is crucial in various fields, such as physics and engineering. For example, when calculating the speed of an object falling under gravity, the formula involves a square root. If you're designing a roller coaster, you need to ensure that the speed at certain points is within safe limits, which requires understanding how the square root function behaves and how to graph it accurately. This ensures the design is both thrilling and safe!
