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Question 2 (Essay Worth 2 points)

Using [tex]\( f(x) = \log x \)[/tex], what is the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) = \log (x+4) \)[/tex]? Explain your reasoning.

Sagot :

GinnyAnsweridn
To find the [tex]\( x \)[/tex]-intercept of the function [tex]\( g(x) = \log(x + 4) \)[/tex], we need to determine where the function intersects the [tex]\( x \)[/tex]-axis. The [tex]\( x \)[/tex]-intercept occurs where [tex]\( g(x) = 0 \)[/tex].

Let's outline the steps to find this intercept:

1. Set [tex]\( g(x) \)[/tex] to 0: The [tex]\( x \)[/tex]-intercept is found where the output of the function is zero.
[tex]\[ g(x) = 0 \][/tex]

2. Substitute the function into the equation:
[tex]\[ \log(x + 4) = 0 \][/tex]

3. Understand the logarithmic property: The logarithmic function [tex]\(\log(b)\)[/tex] equals zero when [tex]\( b = 1 \)[/tex], because [tex]\(\log(1) = 0\)[/tex]. We use this property to equate the inside of the logarithm to 1.
[tex]\[ x + 4 = 1 \][/tex]

4. Solve for [tex]\( x \)[/tex]: Isolate [tex]\( x \)[/tex] by subtracting 4 from both sides of the equation.
[tex]\[ x + 4 = 1 \][/tex]
[tex]\[ x = 1 - 4 \][/tex]
[tex]\[ x = -3 \][/tex]

Therefore, the [tex]\( x \)[/tex]-intercept of the function [tex]\( g(x) = \log(x + 4) \)[/tex] is [tex]\( x = -3 \)[/tex]. This means that the graph of [tex]\( g(x) \)[/tex] intersects the [tex]\( x \)[/tex]-axis at the point [tex]\((-3, 0)\)[/tex].
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