Questions: For f(x)=√x and g(x)=x-2, find the following functions. a. (f ∘ g)(x) ; b. (g ∘ f)(x) ; c. (f ∘ g)(6) ; d. (g ∘ f)(6) a. (f ∘ g)(x)=

Transcript text: For $f(x)=\sqrt{x}$ and $g(x)=x-2$, find the following functions. a. $(f \circ g)(x) ;$ b. $(g \circ f)(x) ;$ c. $(f \circ g)(6) ; d .(g \circ f)(6)$ a. $(\mathrm{f} \circ \mathrm{g})(\mathrm{x})=$ $\square$

Solution

Solution Steps

Step 1: Compute the Composite Functions

For $(f \circ g)(x)$, we substitute $g(x)$ into $f(x)$: $f(g(x)) = \sqrt{x - 2}$, which simplifies to $\sqrt{x - 2}$. For $(g \circ f)(x)$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = \sqrt{x} - 2$, which simplifies to $\sqrt{x} - 2$.

Step 2: Simplify the Resulting Expression

The simplified form of $(f \circ g)(x)$ is $\sqrt{x - 2}$. The simplified form of $(g \circ f)(x)$ is $\sqrt{x} - 2$.

Step 3: Determine the Domain

The domain of $(f \circ g)(x)$ is $\left\{2\right\}$. The domain of $(g \circ f)(x)$ is $\left\{4\right\}$.

Step 4: Evaluate the Composite Functions at x_value

At $x = 6$, $(f \circ g)(x)$ evaluates to $2$. At $x = 6$, $(g \circ f)(x)$ evaluates to $0.45$.

Final Answer:

The simplified forms of the composite functions are $(f \circ g)(x) = \sqrt{x - 2}$ and $(g \circ f)(x) = \sqrt{x} - 2$, with their domains being $\left\{2\right\}$ and $\left\{4\right\}$ respectively. At $x = 6$, $(f \circ g)(x)$ evaluates to $2$ and $(g \circ f)(x)$ evaluates to $0.45$.

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