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The range of a function refers to all the possible output values that a function can produce. For the function \( f(D) = \frac{D^2 + 8D - 8}{(D+4)(D-2)} \), this means identifying all the values of \( f(D) \) that can be achieved as \( D \) varies. When you use a graphing calculator to observe the function's graph, you look at the vertical span of the function's curve.

• If the curve covers an area without vertical gaps, then those values are part of the range.
• If it approaches a line but never touches or crosses it, that line could indicate a gap in the range.

In this function, the curve approaches a horizontal line \( y = 1 \) without ever touching it. This gives a hint that the function's range is all real numbers except \( y = 1 \). Thus, no output value of the function will be precisely one.

A horizontal asymptote is a horizontal line that a graph approaches as the input values of the function become very large in either the positive or negative direction. For a rational function like \( f(D) = \frac{D^2 + 8D - 8}{(D+4)(D-2)} \), determining the horizontal asymptote involves analyzing the degrees of the numerator and the denominator.
The degrees of both numerator and denominator here are 2 (because of the \( D^2 \) terms), so the horizontal asymptote is found by dividing the leading coefficients of these terms, yielding \( y = 1 \).

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, it is the ratio of the leading coefficients.
• If the numerator’s degree is greater, there is no horizontal asymptote.

Thus, for this function, as \( D \to \pm \infty \), \( f(D) \to 1 \). The graph gets closer and closer to \( y = 1 \) but never actually reaches it in those extreme values.

The domain of a function is the set of all possible input values (denoted by \( D \) here) that the function can accept. To find the domain of \( f(D) = \frac{D^2 + 8D - 8}{(D+4)(D-2)} \), we need to determine where the function is undefined. This occurs when the denominator equals zero because division by zero is undefined.

• Set the denominator denominator \((D+4)(D-2)\) to zero.
• Solve, getting \( D = -4 \) and \( D = 2 \).

So, \( D \) cannot be \(-4\) or \(2\). Therefore, the domain of \( f \) is all real numbers except \( D = -4 \) and \( D = 2 \). The function can be evaluated at any other real number value for \( D \).

Vertical asymptotes are lines where the function approaches infinity. They occur at specific \( D \) values where the denominator of a rational function equals zero if those points do not also cancel out in the numerator. For \( f(D) = \frac{D^2 + 8D - 8}{(D+4)(D-2)} \), this means checking where the denominator \((D+4)(D-2)\) becomes zero.

• Solve \((D+4)(D-2) = 0\), giving \( D = -4 \) and \( D = 2 \).
• Check if these values cancel out in the numerator. They do not.

Thus, the vertical asymptotes are located at \( D = -4 \) and \( D = 2 \). This is where the function spikes sharply upwards or downwards, pointing to behavior towards infinity. When you graph the function on a graphing calculator, near these values, the function will demonstrate this trending towards infinity or negative infinity behavior.