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Section 1.1 : Review : Functions

22. Find the domain and range of \(M\left( x \right) = 5 - \left| {x + 8} \right|\).

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We’re dealing with an absolute value here and the quantity inside is a line, which we can plug all values of \(x\) into, and so the domain is all real numbers or,

\[{\mbox{Domain : }} - \infty < x < \infty \,\,\,{\rm{or}}\,\,\,\left( { - \infty ,\infty } \right)\]

For the range let’s again note that the quantity inside the absolute value is a linear function that will take on all real values. We also know that absolute value functions will never be negative and will only be zero if we take the absolute value of zero. So, we now know that,

\[\left| {x + 8} \right| \ge 0\]

However, we are subtracting this from 5 and so we’ll be subtracting a positive or zero number from 5 and so the range is,

\[{\mbox{Range : }}\left( { - \infty ,5} \right]\]