Question

Find $g\left[h\left(x\right)\right] \text{and}h\left[g\left(x\right)\right]$for the functions $h\left(x\right)=2x+5\text{and}g\left(x\right)=-x+3$.

Short answer

The value of $g\left[h\left(x\right)\right]=-2x-2\text{and}h\left[g\left(x\right)\right]=-2x+11$.

Step-by-step solution

Step 1. Write down the given equation.

The given functions are $h\left(x\right)=2x+5\text{and}g\left(x\right)=-x+3$.

Step 2. Evaluating ghx and hgx.

The value of $g\left[h\left(x\right)\right]\text{and}h\left[g\left(x\right)\right]$is evaluated as:

$\begin{array}{c}g\left(x\right)=-x+\mathrm{3....}\left(\text{Given}\right)\\ g\left[h\left(x\right)\right]=-\left[h\left(x\right)\right]+\mathrm{3....}\left(\text{Plugging}x=h\left(x\right)\right)\\ g\left[h\left(x\right)\right]=-\left[2x+5\right]+\mathrm{3....}\left(\because h\left(x\right)=2x+5\right)\\ g\left[h\left(x\right)\right]=-2x-2\\ \text{Now,}h\left(x\right)=2x+\mathrm{5....}\left(\text{Given}\right)\\ h\left[g\left(x\right)\right]=2\left[g\left(x\right)\right]+\mathrm{5....}\left(\text{Plugging}x=g\left(x\right)\right)\\ h\left[g\left(x\right)\right]=2\left(-x+3\right)+\mathrm{5....}\left(\because g\left(x\right)=-x+3\right)\\ h\left[g\left(x\right)\right]=-2x+11\end{array}$

Step 3. Conclusion.

The value of $g\left[h\left(x\right)\right]=-2x-2\text{and}h\left[g\left(x\right)\right]=-2x+11$.