Let's solve the problem step-by-step.
### (a) Graphing the Function and Finding the Zeros
The function given is:
[tex]\[ f(x) = \frac{6x - 1}{x - 5} \][/tex]
A "zero" of the function [tex]\( f(x) \)[/tex] is any value of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex]. This means we need to solve for [tex]\( x \)[/tex] where the numerator is zero, since a fraction is zero only when its numerator is zero (provided the denominator is not zero).
Let's set the numerator to zero:
[tex]\[ 6x - 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 6x = 1 \][/tex]
[tex]\[ x = \frac{1}{6} \approx 0.1667 \][/tex]
So, the zero of the function, rounded to two decimal places, is:
[tex]\[ x = 0.17 \][/tex]
### (b) Verifying the Results Algebraically
We have already found the zero of the function by solving the numerator equal to zero:
[tex]\[ 6x - 1 = 0 \][/tex]
[tex]\[ 6x = 1 \][/tex]
[tex]\[ x = \frac{1}{6} \][/tex]
This confirms that the zero is:
[tex]\[ x = 0.17 \][/tex]
Thus, verifying our result algebraically gives the same solution.
To summarize:
(a) The zero of the function is:
[tex]\[ x = 0.17 \][/tex]
(b) Verifying this result algebraically confirms:
[tex]\[ x = 0.17 \][/tex]
