Using $f(x)=\log x$, what is the $x$-intercept of $g(x)=\log (x+4)$? Explain your reasoning.
Answer (2)
Set g ( x ) = 0 to find the x-intercept: lo g ( x + 4 ) = 0 .
Convert the logarithmic equation to exponential form: x + 4 = 1 0 0 = 1 .
Solve for x : x = 1 − 4 = − 3 .
The x-intercept of g ( x ) is − 3 .
Explanation
Understanding the Problem We are given the function f ( x ) = lo g x and g ( x ) = lo g ( x + 4 ) . We need to find the x -intercept of g ( x ) . The x -intercept is the point where the graph of the function intersects the x -axis, which means 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 Assuming the logarithm is base 10, we can rewrite the equation in exponential form: x + 4 = 1 0 0
Simplifying the Equation Since 1 0 0 = 1 , we have: x + 4 = 1
Solving for x Now, we solve for x :
x = 1 − 4 x = − 3
Finding the x-intercept Therefore, the x -intercept of g ( x ) = lo g ( x + 4 ) is x = − 3 . The x -intercept is the point ( − 3 , 0 ) . The reasoning is that the x -intercept occurs where the function's value is zero. By setting g ( x ) to zero and solving for x , we find the x -coordinate of the x -intercept.
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 the company's revenue equals its costs. Similarly, in physics, the x-intercept of a projectile's trajectory can indicate the point where it lands. Knowing how to find and interpret x-intercepts allows us to analyze and make informed decisions in various fields.
The x-intercept of g ( x ) = lo g ( x + 4 ) is found by setting the function equal to zero, leading to the equation x + 4 = 1 . Solving for x gives x = − 3 , which means the function intersects the x-axis at the point ( − 3 , 0 ) .
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