Problem 15 - College Algebra Essentials

Question

Find the domain of each function. $$f(x)=\frac{1}{\frac{4}{x-1}-2}$$

Step-by-Step Solution

Answer

The domain of the function \( f(x) = \frac{1}{\frac{4}{x-1}-2} \) are all real numbers excluding 1 and 3.

Step 1: Identify values of x that make the inner denominator zero

Examine the inner fraction in the denominator of the function. The denominator of this fraction is \( x-1 \), and it would be zero when \( x=1 \). Therefore, \( x=1 \) is excluded from the domain because it would make the denominator undefined.

Step 2: Identify values of x that make the overall denominator zero

Investigate the overall denominator of the function, which is \( \frac{4}{x-1}-2 \). This is zero when \( \frac{4}{x-1} = 2 \). Solving this equation for x gives \( x=\frac{4}{2}+1 = 3 \). Therefore, \( x=3 \) is excluded from the domain because it makes the overall denominator zero.

Step 3: Construct the domain

From steps 1 and 2, we have established that the domain excludes the values \( x=1 \) and \( x=3 \). Hence, the domain of function f(x) consists of all real numbers except 1 and 3.