Angelica is working on function machines. She has two machines g(x)=square root x-5 and h(x)= x^2-6. she wants to put them in order so that the output of the first machine becomes the input of the second. she wants to use a beginning input of 6.

Ain what order must she put the machines to get a final output of 5.

b)is it possible for her to get a final output of -5? if so,show how she could do that. If not explain why not.

PLEASE HELP!

\(g(h(x))=\sqrt{x^2-6-5}=\sqrt{x^2-11}\\ g(h(6))=\sqrt{6^2-11}=\sqrt{36-11}=\sqrt{25}=5\)

a)
\(h(x)\) is the input for \(g(x)\), so \(h(x)\) must be first

b)
It's impossible for \(g(h(x)=\sqrt{x^2-11}\), because its value is always non-negative for any x. Let's see what about \(h(g(x))\).

\(h(g(x))=(\sqrt{x-5})^2-6=x-5-6=x-11\)

The result is a non-constant linear function, so its value can be any real number, including -5. You can calculate for what x it's equal to -5.

\(x-11=-5\\ x=6\)

\(x-11=-5\\ x=6\)

johanrusli johanrusli · Answer 2 · 2019-08-08T03:23:04-04:00

a) To get a final output of 5 , she must first input 6 into machine h(x) , then the result from machine h(x) is input back to machine g(x).

b) It is possible to get a final output of -5. It could be done by first input 6 into machine g(x) , then the result from machine g(x) is input back to machine h(x).

Further explanation

Function is a relation which each member of the domain is mapped onto exactly one member of the codomain.

There are many types of functions in mathematics such as :

Linear Function → f(x) = ax + b

Quadratic Function → f(x) = ax² + bx + c
Trigonometric Function → f(x) = sin x or f(x) = cos x or f(x) = tan x
Logarithmic function → f(x) = ln x
Polynomial function → f(x) = axⁿ + bxⁿ⁻¹ + ...

If function f : x → y , then inverse function f⁻¹ : y → x

Let us now tackle the problem!

This problem is about Composition of Functions.

Question a:

Given:

\(g(x) = \sqrt{x - 5}\)

\(h(x) = x^2 - 6\)

\(( h \circ g )( x ) = h ( g ( x ) )\)

\(( h \circ g )( x ) = h ( \sqrt {x - 5} )\)

\(( h \circ g )( x ) = (\sqrt {x - 5})^2 - 6\)

\(( h \circ g )( x ) = x - 5 - 6\)

\(( h \circ g )( x ) = x - 11\)

\(( h \circ g )( 6 ) = 6 - 11\)

\(\large {\boxed {( h \circ g )( 6 ) = -5 } }\)

\(( g \circ h )( x ) = g ( h ( x ) )\)

\(( g \circ h )( x ) = g ( x^2 - 6 )\)

\(( g \circ h )( x ) = \sqrt {( x^2 - 6 ) - 5 }\)

\(( g \circ h )( x ) = \sqrt { x^2 - 11 }\)

\(( g \circ h )( 6 ) = \sqrt { 6^2 - 11 }\)

\(( g \circ h )( 6 ) = \sqrt { 25 }\)

\(\large {\boxed {( g \circ h )( 6 ) = 5 } }\)

To get a final output of 5 , she must first input 6 into machine h(x) , then the result from machine h(x) is input back to machine g(x).

Question b:

From the results above , it is possible to get a final output of -5.

It could be done by first input 6 into machine g(x) , then the result from machine g(x) is input back to machine h(x).

Learn more

Inverse of Function : https://brainly.com/question/9289171
Rate of Change : https://brainly.com/question/11919986
Graph of Function : https://brainly.com/question/7829758

Answer details

Grade: High School

Subject: Mathematics

Chapter: Function

Keywords: Function , Trigonometric , Linear , Quadratic