Answer :
To determine the value of [tex]\( f(-1/7) \)[/tex] for the function [tex]\( f(x) = \frac{4x - 3}{1 + 7x} \)[/tex], we need to follow these steps.
1. Substitute [tex]\( x \)[/tex] with [tex]\(-\frac{1}{7}\)[/tex] in the function [tex]\( f(x) \)[/tex].
[tex]\[ f\left(-\frac{1}{7}\right) = \frac{4\left(-\frac{1}{7}\right) - 3}{1 + 7\left(-\frac{1}{7}\right)} \][/tex]
2. Simplify the numerator:
[tex]\[ 4\left(-\frac{1}{7}\right) - 3 = -\frac{4}{7} - 3 \][/tex]
Convert -3 into a fraction with a common denominator:
[tex]\[ -\frac{4}{7} - 3 = -\frac{4}{7} - \frac{21}{7} = -\frac{25}{7} \][/tex]
3. Simplify the denominator:
[tex]\[ 1 + 7\left(-\frac{1}{7}\right) = 1 - 1 = 0 \][/tex]
4. Compute the value of the function:
[tex]\[ f\left(-\frac{1}{7}\right) = \frac{-\frac{25}{7}}{0} \][/tex]
Since the denominator is zero, the value is undefined, which means it approaches infinity.
Therefore, the correct answer is:
[tex]\[ \boxed{\infty} \][/tex]
1. Substitute [tex]\( x \)[/tex] with [tex]\(-\frac{1}{7}\)[/tex] in the function [tex]\( f(x) \)[/tex].
[tex]\[ f\left(-\frac{1}{7}\right) = \frac{4\left(-\frac{1}{7}\right) - 3}{1 + 7\left(-\frac{1}{7}\right)} \][/tex]
2. Simplify the numerator:
[tex]\[ 4\left(-\frac{1}{7}\right) - 3 = -\frac{4}{7} - 3 \][/tex]
Convert -3 into a fraction with a common denominator:
[tex]\[ -\frac{4}{7} - 3 = -\frac{4}{7} - \frac{21}{7} = -\frac{25}{7} \][/tex]
3. Simplify the denominator:
[tex]\[ 1 + 7\left(-\frac{1}{7}\right) = 1 - 1 = 0 \][/tex]
4. Compute the value of the function:
[tex]\[ f\left(-\frac{1}{7}\right) = \frac{-\frac{25}{7}}{0} \][/tex]
Since the denominator is zero, the value is undefined, which means it approaches infinity.
Therefore, the correct answer is:
[tex]\[ \boxed{\infty} \][/tex]
