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Chapter 9: Problem 174

If \(\mathrm{f}(\mathrm{x})=3 \mathrm{x}+4\) and \(\mathrm{D}=\\{\mathrm{x} \mid-1 \leq \mathrm{x} \leq 3\\}\), find the range of \(\mathrm{f}(\mathrm{x})\)

Short Answer

Expert verified
The range of \(f(x) = 3x + 4\) within the domain \(\{x \mid -1 \leq x \leq 3\}\) is \(\{f(x) \mid 1 \leq f(x) \leq 13\}\).

Step by step solution

01

Identify the function and the domain of interest.

We are given a function \(f(x) = 3x + 4\) and a domain \(D = \{x \mid -1 \leq x \leq 3\}\).
02

Find the minimum and maximum values of the function within the given domain.

To find the minimum and maximum values of \(f(x)\), we evaluate the function at the endpoints of the domain: For the lower endpoint, \(x = -1\): \(f(-1) = 3(-1) + 4 = -3 + 4 = 1\) For the upper endpoint, \(x = 3\): \(f(3) = 3(3) + 4 = 9 + 4 = 13\) Since the function is a straight line with a positive slope, it will continuously increase from the lower endpoint value to the upper endpoint value.
03

Find the range of the function within the given domain.

Now that we have found the minimum and maximum values of \(f(x)\) within the given domain, we can express the range as the set of all possible values between these endpoints: Range of \(f(x) = \{f(x) \mid 1 \leq f(x) \leq 13\}\)

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