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Horizontal asymptotes are lines that a function approaches as the input, or \( x \), grows infinitely large or goes to negative infinity. They tell us the behavior of the function at the extreme ends of the \( x \)-axis. To find horizontal asymptotes, we often look at the term with the highest power of \( x \) in a rational function.
For example, consider the function \( y = 4 + \frac{3x^2}{x^2+1} \). When \( x \) becomes very large (whether positively or negatively), the fraction \( \frac{3x^2}{x^2+1} \) simplifies to \( 3 \) because \( x^2+1 \) is nearly \( x^2 \) itself, making \( \frac{3x^2}{x^2+1} \approx 3 \).
The horizontal asymptote of the function is found by adding this simplified value to the constant term in the function. Thus, for this function, we get a horizontal asymptote at \( y = 4 + 3 = 7 \).

• Horizontal asymptotes describe end behavior, not necessarily bounds on the range.
• They are not affected by finite changes, only by behavior at extreme values of \( x \).

Vertical asymptotes are lines \( x = a \) where a function's value increases or decreases indefinitely. These occur when the denominator of a fraction approaches zero, but the numerator does not also become zero simultaneously.
In the function \( y = 4 + \frac{3x^2}{x^2+1} \), the denominator \( x^2+1 \) never equals zero for any real \( x \) since \( x^2+1 \geq 1 \). Therefore, there are no vertical asymptotes in this function.

• Vertical asymptotes indicate values where a function becomes undefined.
• They can be found by setting the denominator equal to zero unless the numerator also zeroes out at the same points (which would be holes).

The range of a function is the set of possible output values. To determine the range, we look at the behavior of the function over all possible inputs.
For the function \( y = 4 + \frac{3x^2}{x^2+1} \), the expression \( \frac{3x^2}{x^2+1} \) gives us clues about the range. This component has a minimum value of \( 0 \) when \( x = 0 \) and a maximum value of \( 3 \) as \( x \) approaches infinity.
Adding the constant \( 4 \) to both the minimum and maximum values, the overall range of the function becomes \([4, 7)\). This means the function can take on values starting from \( 4 \) up to, but not including, \( 7 \).

• Finding the range involves determining the lowest and highest values the function can achieve.
• The range can be affected by asymptotic behavior and constants in the function.