Questions: Find (f ∘ g)(x) and (g ∘ f)(x). f(x)=6x+1 ; g(x)=√x (f ∘ g)(x)=

Transcript text: Find $(f \circ g)(x)$ and $(g \circ f)(x)$. \[ f(x)=6 x+1 ; \quad g(x)=\sqrt{x} \] $(f \circ g)(x)=$ $\square$

Solution

Solution Steps

To find $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$. Similarly, to find $(g \circ f)(x)$, we substitute $f(x)$ into $g(x)$. This involves function composition, where one function is applied to the result of another function.

Step 1: Find $ (f \circ g)(x) $

To find $ (f \circ g)(x) $, we substitute $ g(x) = \sqrt{x} $ into $ f(x) = 6x + 1 $: \[ (f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = 6\sqrt{x} + 1 \]

Step 2: Find $ (g \circ f)(x) $

Next, we find $ (g \circ f)(x) $ by substituting $ f(x) = 6x + 1 $ into $ g(x) = \sqrt{x} $: \[ (g \circ f)(x) = g(f(x)) = g(6x + 1) = \sqrt{6x + 1} \]

Final Answer

Thus, the results are: \[ (f \circ g)(x) = 6\sqrt{x} + 1 \] \[ (g \circ f)(x) = \sqrt{6x + 1} \] The final answers are: \[ \boxed{(f \circ g)(x) = 6\sqrt{x} + 1} \] \[ \boxed{(g \circ f)(x) = \sqrt{6x + 1}} \]

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