Questions: What is the range of the function (f(x)=4x-5) over the interval of (1<x leq 5) ? Give your answer in interval notation.

What is the range of the function (f(x)=4x-5) over the interval of (1<x leq 5) ? Give your answer in interval notation.
Transcript text: Type your answer (11 out of 20) What is the range of the function $f(x)=4 x-5$ over the interval of $1
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Solution

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Solution Steps

Step 1: Determine the Function Values at the Interval Endpoints

The function given is \( f(x) = 4x - 5 \). We need to evaluate this function at the endpoints of the interval \( 1 < x \leq 5 \).

  • Evaluate \( f(x) \) at \( x = 1 \): \[ f(1) = 4(1) - 5 = 4 - 5 = -1 \]

  • Evaluate \( f(x) \) at \( x = 5 \): \[ f(5) = 4(5) - 5 = 20 - 5 = 15 \]

Step 2: Determine the Range of the Function

Since the function \( f(x) = 4x - 5 \) is a linear function with a positive slope, it is increasing over the interval \( 1 < x \leq 5 \). Therefore, the range of \( f(x) \) will be from just above \( f(1) \) to \( f(5) \).

  • The lower bound of the range is just above \( f(1) = -1 \), but since \( x = 1 \) is not included in the interval, the range starts from a value slightly greater than \(-1\).
  • The upper bound of the range is \( f(5) = 15 \).

Thus, the range of the function over the interval \( 1 < x \leq 5 \) is \((-1, 15]\).

Final Answer

\[ \boxed{(-1, 15]} \]

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