Verified Answer :

BladeRunner212

g(h(-3)) = ⁸/₅ = 1.6

Further explanation  

if g(x)=x+1/x-2 and h(x) = 4 – x, what is the value of g(h(-3))

In this problem we will find out the value of the function composition. There are two ways to do it.  

\(\boxed{ \ g(x) = \frac{x + 1}{x - 2} \ }\)  

\(\boxed{ \ h(x) = 4 - x \ }\)  

\(\boxed{ \ g(h(-3)) = ? \ }\)  

Option 1

Step-1: compose (g o h)(x) = g(h(x))  

Here h(x) as input into g(x). In other words, first we apply h(x), then apply g(x) to that result:  

\(h(x) = 4 - x \rightarrow g(x) = \frac{x + 1}{x - 2}\)  

\(g(h(x)) = \frac{(4 - x) + 1}{(4 - x) - 2}\)  

And we get,  

\(\boxed{ \ g(h(x)) = \frac{5 - x}{2 - x} \ }\)  

Step-2: calculate the value of g(h(-3))  

After getting g(h(x)) we proceed by calculating the value g(h(-3)).

\(x = -3 \rightarrow g(h(x)) = \frac{5 - x}{2 - x}\)  

\(g(h(-3)) = \frac{5 - (-3)}{2 - (-3)}\)

\(g(h(-3)) = \frac{5 + 3}{2 + 3}\)  

And we obtain the final result:  

\(\boxed{\boxed{ \ g(h(-3)) = \frac{8}{5} = 1.6 \ }}\)  

Option 2  

Step-1: count h(-3) initially  

\(x = -3 \rightarrow h(-3) = 4 - (-3)\)  

And we get,

\(\boxed{ \ h(-3) = 7 \ }\)  

Step-2: calculate the value of g(h(-3))  

Here the value of h(-3), i.e. 7, as input into g(x).  

\(h(-3) = 7 \rightarrow f(7) = \frac{7 + 1}{7 - 2}\)

\(f(7) = \frac{8}{5}\)

Remember, f(7) as g(h(-3))

And we obtain the final result:  

\(\boxed{\boxed{ \ g(h(-3)) = \frac{8}{5} = 1.6 \ }}\)

Learn more  

  1. Let f(x) = x-3 and g(x ) = x^2 find f(g(4)) https://brainly.com/question/1052893  
  2. The composite function https://brainly.com/question/1691598
  3. Let f(x) = x + 8 and g(x) = x2 - 6x - 7. Find f(g(2)) https://brainly.com/question/2142762

Keywords: if g(x) = x+1/x-2 and h(x) = 4 – x, what is the value of g(h(-3)), composition function, input, g(h(x)), value, initially,  f(7), h(-3)

View image BladeRunner212

The value of \(g\left(h(-3)\right)\) is \(\fbox{\begin\\\ \bf \dfrac{8}{5}\\\end{minispace}}\)

Further explanation:

Given:

The functions are \(g(x)=\frac{x+1}{x-2}\) and \(h(x)=4-x\).

Calculation:

A composite function is defined as a function which comprises of two function in which one function act as an independent variable for the other function.

The composite function for the function \(f\ \text{and}\ g\) can be defined as,

\(\fbox{\begin\\\ \begin{aligned} \math (f\circ g)&=f(g(x))\\(g\circ f)&=g(f(x))\end{aligned} \\\end{minispace}}\)  

The function \((f\circ g)&=f(g(x))\) is read as " \(f\) composition \(\ g\) " and the function \((g\circ f)=g(f(x))\) can be read as " \(\ g\) composition \(f\) ".

The value of " \(f\) composition \(\ g\) " may or may not be equal to " \(\ g\) composition \(f\) ".

If we put \(\fbox{\begin\\\ \math x=h(x)\\\end{minispace}}\) in the function \(g(x)\) then the obtained expression is composite function \(g(h(x))\) and it is written as follows,

\(\fbox{\begin\\\ g(h(x))=\dfrac{h(x)+1}{h(x)-2}\\\end{minispace}}\)  ....(1)

Now, substitute \((4-x)\) for \(h(x)\) in the right hand side of the equation (1) as follows:

\(\begin{aligned}g(h(x))&=\dfrac{(4-x)+1}{(4-x)-2}\\&=\dfrac{(4+1)-x}{(4-2)-x}\\&=\dfrac{5-x}{2-x}\end{aligned}\)

Therefore, the composite function \(g(h(x))\) is \(\frac{5-x}{2-x}\).

Now, to obtain the value of composite function \(g(h(x))\) at \(x=-3\), proceed as shown below.

Substitute \((-3)\) for \(x\) in the above equation as follows,

\(\begin{aligned}g(h(-3))=\dfrac{5-(-3)}{2-(-3)}\\=\dfrac{5+3}{2+3}\\=\dfrac{8}{5}\end{aligned}\)

Therefore, the value of \(g(h(-3))\) is \(\fbox{\begin\\\ \bf \dfrac{8}{5}\\\end{minispace}}\)

Learn more:

1. A problem on functions https://brainly.com/question/2142762

2. A problem on triangle https://brainly.com/question/2992432.

3. A problem on circle https://brainly.com/question/1506955

Answer details

Grade: Senior school

Subject: Mathematics

Chapter: Function

Keywords: g(x)=(x+1)/(x-2), function, composite function, (fog)(x)=f(g(x)), f composition g, g composition f, value of composite function, deppendent variable, independent variable.

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