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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.  

[tex]\boxed{ \ g(x) = \frac{x + 1}{x - 2} \ }[/tex]  

[tex]\boxed{ \ h(x) = 4 - x \ }[/tex]  

[tex]\boxed{ \ g(h(-3)) = ? \ }[/tex]  

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:  

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

[tex]g(h(x)) = \frac{(4 - x) + 1}{(4 - x) - 2}[/tex]  

And we get,  

[tex]\boxed{ \ g(h(x)) = \frac{5 - x}{2 - x} \ }[/tex]  

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

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

[tex] x = -3 \rightarrow g(h(x)) = \frac{5 - x}{2 - x} [/tex]  

[tex] g(h(-3)) = \frac{5 - (-3)}{2 - (-3)} [/tex]

[tex] g(h(-3)) = \frac{5 + 3}{2 + 3} [/tex]  

And we obtain the final result:  

[tex]\boxed{\boxed{ \ g(h(-3)) = \frac{8}{5} = 1.6 \ }}[/tex]  

Option 2  

Step-1: count h(-3) initially  

[tex] x = -3 \rightarrow h(-3) = 4 - (-3) [/tex]  

And we get,

[tex] \boxed{ \ h(-3) = 7 \ } [/tex]  

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

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

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

[tex] f(7) = \frac{8}{5} [/tex]

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

And we obtain the final result:  

[tex]\boxed{\boxed{ \ g(h(-3)) = \frac{8}{5} = 1.6 \ }}[/tex]

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)

Ver imagen BladeRunner212

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

Further explanation:

Given:

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

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 [tex]f\ \text{and}\ g[/tex] can be defined as,

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

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

The value of " [tex]f[/tex] composition [tex]\ g[/tex] " may or may not be equal to " [tex]\ g[/tex] composition [tex]f[/tex] ".

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

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

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

[tex]\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}[/tex]

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

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

Substitute [tex](-3)[/tex] for [tex]x[/tex] in the above equation as follows,

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

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

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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