Question

if $f(x) = |x| + 9$ and $g(x) = -6$, which describes the range of $(f + g)(x)$?
○ $(f + g)(x) \geq 3$ for all values of $x$
○ $(f + g)(x) \leq 3$ for all values of $x$
○ $(f + g)(x) \leq 6$ for all values of $x$
○ $(f + g)(x) \geq 6$ for all values of $x$

Explanation:

Step1: Find (f + g)(x)

To find \((f + g)(x)\), we add the functions \(f(x)\) and \(g(x)\). Given \(f(x)=\vert x\vert + 9\) and \(g(x)=-6\), we have:

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Step2: Analyze the range of \(\vert x\vert\)

The absolute value function \(\vert x\vert\) has the property that \(\vert x\vert\geq0\) for all real numbers \(x\). This is because the absolute value of a number represents its distance from zero on the number line, and distance is always non - negative.

Step3: Analyze the range of \((f + g)(x)\)

Since \((f + g)(x)=\vert x\vert+ 3\) and \(\vert x\vert\geq0\), we can find the range of \((f + g)(x)\) by adding 3 to both sides of the inequality \(\vert x\vert\geq0\).
If we add 3 to both sides of \(\vert x\vert\geq0\), we get \(\vert x\vert+ 3\geq0 + 3\), which simplifies to \((f + g)(x)=\vert x\vert+ 3\geq3\) for all values of \(x\).

Answer:

\((f + g)(x)\geq3\) for all values of \(x\)