Question

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

Answer 1

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)

Answer 2

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.