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Chapter 3: Problem 40

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)=2 x^{2}-8 x-3$$

Short Answer

Expert verified
The function has a minimum value. This minimum value is -7 and it occurs at x = 2. The domain is all real numbers or \(-\infty, \infty\). And the range is \([-7, \infty)\).

Step by step solution

01

Determining Minimum or Maximum

Looking at the equation \(f(x) = 2x^2 - 8x -3\), the leading coefficient \(a\) is 2, which is greater than zero. So, the function has a minimum value.
02

Locate the Minimum Value

The x-coordinate of the vertex can be found by \(-b / 2a = -(-8) / (2*2) = 2\). To find the y-coordinate, we put the x-value into the function: \(f(2) = 2*(2)^2 - 8*2 -3 = -7\). Hence, the minimum value of the function is -7, which occurs at x = 2.
03

Identify the Domain and Range

The domain of a quadratic function is all real numbers, which can be represented as \(-\infty, \infty\). Given that the function has a minimum value at \(y = -7\), the range of functions is \([-7, \infty)\) as y can take any value greater than or equal to -7.

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