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If [tex]f(x)=\frac{x-3}{x}[/tex] and [tex]g(x)=5x-4[/tex], what is the domain of [tex](f \circ g)(x)[/tex]?

A. [tex]\{x \mid x \neq 0\}[/tex]
B. [tex]\left\{x \mid x \neq \frac{1}{3}\right\}[/tex]
C. [tex]\left\{x \mid x \neq \frac{4}{5}\right\}[/tex]
D. [tex]\{x \mid x \neq 3\}[/tex]


Sagot :

To determine the domain of [tex]\( (f \circ g)(x) \)[/tex], we will follow these steps:

1. Understand the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = \frac{x-3}{x} \)[/tex]
- [tex]\( g(x) = 5x - 4 \)[/tex]

2. Find the composition [tex]\( f(g(x)) \)[/tex]:
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
- This gives us [tex]\( f(g(x)) = f(5x - 4) \)[/tex].
- Replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( 5x - 4 \)[/tex], resulting in [tex]\( f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4} \)[/tex].
- Simplify the numerator: [tex]\( (5x - 4) - 3 = 5x - 7 \)[/tex].
- Therefore, [tex]\( f(g(x)) = \frac{5x - 7}{5x - 4} \)[/tex].

3. Determine conditions for the domain:
- For the composition [tex]\( f(g(x)) \)[/tex] to be defined, we need the denominator to be non-zero.
- The denominator of [tex]\( f(g(x)) \)[/tex] is [tex]\( 5x - 4 \)[/tex].
- So, set [tex]\( 5x - 4 \neq 0 \)[/tex] which ensures the denominator is not zero.
- Solve the inequality: [tex]\( 5x - 4 \neq 0 \)[/tex], leading to [tex]\( x \neq \frac{4}{5} \)[/tex].

Thus, the domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \ne \frac{4}{5} \)[/tex].

Therefore, the correct domain of [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ \left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\} \][/tex]

So, the correct choice is:
[tex]\[ \left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\} \][/tex]