Question

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

Short answer

The value of$g\left[h\left(x\right)\right]=x^2+6x+9\text{and}h\left[g\left(x\right)\right]=x^2+3$.

Step-by-step solution

Step 1. Write down the given equation.

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

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:

$\begin{array}{c}g\left(x\right)=x^2\mathrm{....}\left(\text{Given}\right)\\ g\left[h\left(x\right)\right]={\left[h\left(x\right)\right]}^2\mathrm{....}\left(\text{Plugging}x=h\left(x\right)\right)\\ g\left[h\left(x\right)\right]={\left(x+3\right)}^2\mathrm{....}\left(\because h\left(x\right)=x+3\right)\\ g\left[h\left(x\right)\right]=x^2+6x+9\\ \text{Now,}h\left(x\right)=x+\mathrm{3....}\left(\text{Given}\right)\\ h\left[g\left(x\right)\right]=\left[g\left(x\right)\right]+\mathrm{3....}\left(\text{Plugging}x=g\left(x\right)\right)\\ h\left[g\left(x\right)\right]=\left(x^2\right)+\mathrm{3....}\left(\because g\left(x\right)=x^2\right)\\ h\left[g\left(x\right)\right]=x^2+3\end{array}$

Step 3. Conclusion.

The value of$g\left[h\left(x\right)\right]=x^2+6x+9\text{and}h\left[g\left(x\right)\right]=x^2+3$