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Functions [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions and are defined for all real numbers. Determine the value of each function composition.

[tex]\[
\begin{array}{l}
(h \circ k)(3) = \square \\
(k \circ h)(-4b) = \square
\end{array}
\][/tex]

Sagot :

GinnyAnswer
Given that functions [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, we have the following properties:

1. [tex]\( h \circ k \)[/tex] means [tex]\( h \)[/tex] composed with [tex]\( k \)[/tex], or [tex]\( h(k(x)) = x \)[/tex].
2. [tex]\( k \circ h \)[/tex] means [tex]\( k \)[/tex] composed with [tex]\( h \)[/tex], or [tex]\( k(h(x)) = x \)[/tex].

For the given compositions:

1. [tex]\( (h \circ k)(3) = h(k(3)) \)[/tex].

Since [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, [tex]\( h(k(3)) = 3 \)[/tex].

So, [tex]\( (h \circ k)(3) = 3 \)[/tex].

2. [tex]\( (k \circ h)(-4b) = k(h(-4b)) \)[/tex].

Again, because [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, [tex]\( k(h(-4b)) = -4b \)[/tex].

Thus, [tex]\( (k \circ h)(-4b) = -16 \)[/tex].

Filling in the blanks, we get:

[tex]\[ \begin{array}{l} (h \circ k)(3) = 3 \\ (k \circ h)(-4b) = 16 \\ \end{array} \][/tex]
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