Question

Find​ (a) ​(fcircle​g)(x) and​ (b) ​(gcircle​f)(x).
​f(x)equalsStartRoot x minus 4 EndRoot​; ​g(x)equals5 x minus 6
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Part 1
​(a) ​(fcircle​g)(x)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \((f \circ g)(x)\), which means \(f(g(x))\), we will substitute \(g(x)\) into \(f(x)\).

Given:
- \(f(x) = \sqrt{x - 4}\)
- \(g(x) = 5x - 6\)

Now we need to evaluate \(f(g(x))\):

1. Substitute \(g(x)\) into \(f(x)\):
\[
f(g(x)) = f(5x - 6)
\]
2. Now apply \(g(x)\) to the function \(f(x)\):
\[
f(5x - 6) = \sqrt{(5x - 6) - 4}
\]
3. Simplify inside the square root:
\[
f(5x - 6) = \sqrt{5x - 10}
\]

Therefore, the answer for part (a) is:
\[
\boxed{\sqrt{5x - 10}}
\]

Now for part (b), we need to find \((g \circ f)(x)\), which means \(g(f(x))\).

Following a similar process:

1. Substitute \(f(x)\) into \(g(x)\):
\[
g(f(x)) = g(\sqrt{x - 4})
\]
2. Now apply \(f(x)\) to the function \(g(x)\):
\[
g(\sqrt{x - 4}) = 5(\sqrt{x - 4}) - 6
\]

Thus, the result for part (b) is:
\[
\boxed{5\sqrt{x - 4} - 6}
\]