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Determining whether two functions are inverses of each other please help

Determining whether two functions are inverses of each other please help-example-1
User Pgilmon
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2 Answers

3 votes

Answer:

see explanation

Explanation:

given f(x) and g(x)

if f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

(a)

f(g(x))

= f(- 2x)

= -
(-2x)/(2) ( cancel 2 on numerator/ denominator )

= x

g(f(x))

= g(-
(x)/(2) )

= - 2 × -
(x)/(2) ( cancel 2 on numerator/ denominator )

= x

since f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

(b)

f(g(x))

= f(
(x-1)/(2) )

= 2(
(x-1)/(2) ) + 1

= x - 1 + 1

= x

g(f(x))

= g(2x + 1)

=
(2x+1-1)/(2)

=
(2x)/(2)

= x

since f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

User FAHID
by
8.4k points
6 votes

Answer:


\begin{aligned} \textsf{(a)} \quad f(g(x))&=\boxed{x}\\g(f(x))&=\boxed{x}\end{aligned}\\\\\textsf{\;\;\;\;\;\;\;\;$f$ and $g$ are inverses of each other.}


\begin{aligned} \textsf{(b)} \quad f(g(x))&=\boxed{x}\\g(f(x))&=\boxed{x}\end{aligned}\\\\\textsf{\;\;\;\;\;\;\;\;$f$ and $g$ are inverses of each other.}

Explanation:

Part (a)

Given functions:


\begin{cases}f(x)=-(x)/(2)\\\\g(x)=-2x\end{cases}

Evaluate the composite function f(g(x)):


\begin{aligned}f(g(x))&=f(-2x)\\\\&=-(-2x)/(2)\\\\&=x\end{aligned}

Evaluate the composite function g(f(x)):


\begin{aligned}g(f(x))&=g\left(-(x)/(2)\right)\\\\&=-2\left(-(x)/(2)\right)\\\\&=x\end{aligned}

The definition of inverse functions states that two functions, f and g, are inverses of each other if and only if their compositions yield the identity function, i.e. f(g(x)) = g(f(x)) = x.

Therefore, as f(g(x)) = g(f(x)) = x, then f and g are inverses of each other.


\hrulefill

Part (b)

Given functions:


\begin{cases}f(x)=2x+1\\\\g(x)=(x-1)/(2)\end{cases}

Evaluate the composite function f(g(x)):


\begin{aligned}f(g(x))&=f\left((x-1)/(2)\right)\\\\&=2\left((x-1)/(2)\right)+1\\\\&=(x-1)+1\\\\&=x\end{aligned}

Evaluate the composite function g(f(x)):


\begin{aligned}g(f(x))&=g(2x+1)\\\\&=((2x+1)-1)/(2)\\\\&=(2x)/(2)\\\\&=x\end{aligned}

The definition of inverse functions states that two functions, f and g, are inverses of each other if and only if their compositions yield the identity function, i.e. f(g(x)) = g(f(x)) = x.

Therefore, as f(g(x)) = g(f(x)) = x, then f and g are inverses of each other.

User Alecswan
by
9.1k points

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