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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.

a) in 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!

Respuesta :

konrad509
[tex]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[/tex]

a)
[tex]h(x)[/tex] is the input for [tex]g(x)[/tex], so [tex]h(x)[/tex] must be first

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

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

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.

[tex]x-11=-5\\ x=6[/tex]

[tex]x-11=-5\\ x=6[/tex]
johanrusli

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:

[tex]g(x) = \sqrt{x - 5}[/tex]

[tex]h(x) = x^2 - 6[/tex]

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

[tex]( h \circ g )( x ) = h ( \sqrt {x - 5} )[/tex]

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

[tex]( h \circ g )( x ) = x - 5 - 6[/tex]

[tex]( h \circ g )( x ) = x - 11[/tex]

[tex]( h \circ g )( 6 ) = 6 - 11[/tex]

[tex]\large {\boxed {( h \circ g )( 6 ) = -5 } }[/tex]

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

[tex]( g \circ h )( x ) = g ( x^2 - 6 )[/tex]

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

[tex]( g \circ h )( x ) = \sqrt { x^2 - 11 }[/tex]

[tex]( g \circ h )( 6 ) = \sqrt { 6^2 - 11 }[/tex]

[tex]( g \circ h )( 6 ) = \sqrt { 25 }[/tex]

[tex]\large {\boxed {( g \circ h )( 6 ) = 5 } }[/tex]

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

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