Plot the [tex]$x$[/tex]-intercept of the given function.
[tex]$g(x)=\log (x+4)$[/tex]
Answer (2)
Set the function equal to zero: lo g ( x + 4 ) = 0 .
Convert the logarithmic equation to exponential form: 1 0 0 = x + 4 .
Simplify and solve for x : 1 = x + 4 ⇒ x = − 3 .
The x-intercept is − 3 .
Explanation
Understanding the Problem We are given the function g ( x ) = lo g ( x + 4 ) and asked to find its x -intercept. The x -intercept is the point where the graph of the function intersects the x -axis. This occurs when g ( x ) = 0 .
Setting up the Equation To find the x -intercept, we set g ( x ) = 0 and solve for x :
lo g ( x + 4 ) = 0
Converting to Exponential Form Since the base of the logarithm is not specified, we assume it is base 10. Therefore, we can rewrite the equation in exponential form: 1 0 0 = x + 4
Simplifying Simplifying the equation, we have: 1 = x + 4
Solving for x Solving for x , we subtract 4 from both sides: x = 1 − 4 = − 3
Finding the x-intercept Thus, the x -intercept is the point ( − 3 , 0 ) . We plot this point on the graph.
Examples
Understanding x-intercepts is crucial in many real-world applications. For example, in business, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. In physics, it can represent the point where a projectile lands. In environmental science, it can represent the point where a population reaches zero. Knowing how to find and interpret x-intercepts allows us to analyze and make informed decisions in various fields.
The x-intercept of the function g ( x ) = lo g ( x + 4 ) is found by setting g ( x ) = 0 , which gives x = − 3 . Therefore, the x-intercept is the point ( − 3 , 0 ) . This point indicates where the graph intersects the x-axis.
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