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Functions [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are inverse functions, and both are defined for all real numbers.
Using this relationship, what is the value of each function composition?

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



Answer :

GinnyAnswer
To solve these problems, we need to understand the properties of inverse functions. By definition, if [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, then for any value of [tex]\( x \)[/tex]:

[tex]\[ (h \circ k)(x) = x \][/tex]

and

[tex]\[ (k \circ h)(x) = x \][/tex]

This means that when we compose [tex]\( h \)[/tex] and [tex]\( k \)[/tex], or compose [tex]\( k \)[/tex] and [tex]\( h \)[/tex], we should get the input value back.

1. For the first part, [tex]\( (h \circ k)(3) \)[/tex]:

Since [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, applying [tex]\( k \)[/tex] first and then [tex]\( h \)[/tex] should return the original input value. Therefore:

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

2. For the second part, [tex]\( (k \circ h)(-4b) \)[/tex]:

Again, since [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are inverse functions, applying [tex]\( h \)[/tex] first and then [tex]\( k \)[/tex] should return the original input value. Therefore:

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

So, the values are:

[tex]\[ (h \circ k)(3) = 3 \][/tex]
[tex]\[ (k \circ h)(-4b) = -4b \][/tex]

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