Question

Find $g\left[h\left(x\right)\right] \text{and}h\left[g\left(x\right)\right]$for the functions $h\left(x\right)=2x-1 \text{and}g\left(x\right)=x-5$.

Short answer

The value of$g\left[h\left(x\right)\right]=2x-6\text{and}h\left[g\left(x\right)\right]=2x-11$.

Step-by-step solution

Step 1. Write down the given equation.

The given functions are $h\left(x\right)=2x-1\text{and}g\left(x\right)=x-5$.

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:

$\begin{array}{c}g\left(x\right)=x-\mathrm{5....}\left(\text{Given}\right)\\ g\left[h\left(x\right)\right]=\left[h\left(x\right)\right]-\mathrm{5....}\left(\because h\left(x\right)=2x-1\right)\\ g\left[h\left(x\right)\right]=\left(2x-1\right)-5\\ g\left[h\left(x\right)\right]=2x-6\\ \text{Now,}h\left(x\right)=2x-\mathrm{1....}\left(\text{Given}\right)\\ h\left[g\left(x\right)\right]=2\left[g\left(x\right)\right]-\mathrm{1....}\left(\because g\left(x\right)=x-5\right)\\ h\left[g\left(x\right)\right]=2\left(x-5\right)-1\\ h\left[g\left(x\right)\right]=2x-11\end{array}$

Step 3. Conclusion.

The value of $g\left[h\left(x\right)\right]=2x-6\text{and}h\left[g\left(x\right)\right]=2x-11$.