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$x=-1$ $\newline$ $x=1$ $\newline$ $f(x)=1$ $\newline$ $g(x)=-1$ $\newline$ Determine whether pair of function $F$ and $g$ are inverses. Explain. your reasonin.

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Find the slope of a tangent line using limits

Full solution

Q.

x=-1

\newline

x=1

\newline

f(x)=1

\newline

g(x)=-1

\newline

Determine whether pair of function

F

and

g

are inverses. Explain. your reasonin.

Check Composition Results: To determine if two functions are inverses of each other, we need to check if the composition of the two functions results in the identity function. The composition of $f$ and $g$ ( $f(g(x))$ ) should equal $x$ , and the composition of $g$ and $f$ ( $g(f(x))$ ) should also equal $x$ .
Compute $f(g(x))$ : Let's first compute $f(g(x))$ . Since $g(x) = -1$ , we substitute $-1$ for $x$ in the function $f(x)$ . $\newline$ $f(g(x)) = f(-1) = 1$ (since $f(x)$ is given as $1$ for any $x$ ).
Compute $g(f(x))$ : Now let's compute $g(f(x))$ . Since $f(x) = 1$ , we substitute $1$ for $x$ in the function $g(x)$ . $g(f(x)) = g(1) = -1$ (since $g(x)$ is given as $-1$ for any $x$ ).
Evaluate Compositions: We see that $f(g(x)) = 1$ and $g(f(x)) = -1$ . For $f$ and $g$ to be inverses, we would need $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domain of the respective functions. However, this is not the case here.
Final Conclusion: Since $f(g(x)) \neq x$ and $g(f(x)) \neq x$ , we can conclude that $f$ and $g$ are not inverses of each other.

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