To find [tex]\((f-g)(x)\)[/tex] when given the functions [tex]\(f(x) = \frac{2x + 6}{3x}\)[/tex] and [tex]\(g(x) = \frac{\sqrt{x} - 8}{3x}\)[/tex], we need to follow several steps:
1. Evaluate [tex]\(f(x)\)[/tex]:
The function [tex]\(f(x)\)[/tex] is defined as:
[tex]\[
f(x) = \frac{2x + 6}{3x}
\][/tex]
2. Evaluate [tex]\(g(x)\)[/tex]:
The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[
g(x) = \frac{\sqrt{x} - 8}{3x}
\][/tex]
3. Compute [tex]\(f(x)\)[/tex] with a specific value of [tex]\(x\)[/tex]:
Let's choose [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = \frac{2 \cdot 4 + 6}{3 \cdot 4} = \frac{8 + 6}{12} = \frac{14}{12} = \frac{7}{6} \approx 1.1666666666666667
\][/tex]
4. Compute [tex]\(g(x)\)[/tex] with the same value [tex]\(x = 4\)[/tex]:
[tex]\[
g(4) = \frac{\sqrt{4} - 8}{3 \cdot 4} = \frac{2 - 8}{12} = \frac{-6}{12} = -\frac{1}{2} = -0.5
\][/tex]
5. Find [tex]\( (f-g)(x) \)[/tex] using [tex]\(x = 4\)[/tex]:
[tex]\[
(f - g)(4) = f(4) - g(4)
\][/tex]
6. Compute the result:
[tex]\[
(f - g)(4) = \frac{7}{6} - (-\frac{1}{2}) = \frac{7}{6} + \frac{1}{2}
\][/tex]
To add these fractions, we need a common denominator. The least common multiple of 6 and 2 is 6. Convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with a denominator of 6:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{7}{6} + \frac{3}{6} = \frac{10}{6} = \frac{5}{3} \approx 1.6666666666666667
\][/tex]
Therefore, the final results are:
[tex]\[
f(4) = \frac{7}{6} \approx 1.1666666666666667
\][/tex]
[tex]\[
g(4) = -0.5
\][/tex]
[tex]\[
(f - g)(4) = \frac{5}{3} \approx 1.6666666666666667
\][/tex]
In summary:
- [tex]\(f(4) = 1.1666666666666667\)[/tex]
- [tex]\(g(4) = -0.5\)[/tex]
- [tex]\((f - g)(4) = 1.6666666666666667\)[/tex]
