Question

given $f(x) = \frac{4x - 7}{8x + 8}$, what is the end behavior of the function?\
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\bigcirc as $x \to -\infty$, $f(x) \to 0.5$; as $x \to \infty$, $f(x) \to 0.5$.\
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\bigcirc as $x \to -\infty$, $f(x) \to -0.5$; as $x \to \infty$, $f(x) \to -0.5$.\
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\bigcirc as $x \to -\infty$, $f(x) \to 0.73$; as $x \to \infty$, $f(x) \to 0.73$.\
\
\bigcirc as $x \to -\infty$, $f(x) \to -0.73$; as $x \to \infty$, $f(x) \to -0.73$.

Explanation:

Step1: Recall End Behavior of Rational Functions

For a rational function \( f(x) = \frac{a_nx^n + \dots + a_0}{b_mx^m + \dots + b_0} \), if the degrees of the numerator (\( n \)) and denominator (\( m \)) are equal (\( n = m \)), the end behavior is determined by the ratio of the leading coefficients, i.e., \( \frac{a_n}{b_m} \).

Step2: Identify Leading Coefficients

In the function \( f(x) = \frac{4x - 7}{8x + 8} \), the leading term of the numerator is \( 4x \) (leading coefficient \( 4 \)) and the leading term of the denominator is \( 8x \) (leading coefficient \( 8 \)).

Step3: Calculate the Ratio of Leading Coefficients

The ratio of the leading coefficients is \( \frac{4}{8} = \frac{1}{2} = 0.5 \). Since the degrees of the numerator and denominator are both \( 1 \) (equal), as \( x \to \pm\infty \), \( f(x) \) approaches this ratio. So as \( x \to -\infty \), \( f(x) \to 0.5 \) and as \( x \to \infty \), \( f(x) \to 0.5 \).

Answer:

As \( x \to -\infty \), \( f(x) \to 0.5 \); as \( x \to \infty \), \( f(x) \to 0.5 \). (The corresponding option from the choices, e.g., if the first option is "As \( x \to -\infty \), \( f(x) \to 0.5 \); as \( x \to \infty \), \( f(x) \to 0.5 \)", then the answer is that option.)