Question

Find the value of $k\left(0\right),k\left(2\right),k\left(-2\right),k\left(\sqrt{2}\right),k\left(a+2\right),k\left(-x\right),k\left(x^2\right)$for the function $k\left(x\right)=-x^2-2x+3$.

Short answer

The value of function are

$\begin{array}{l}k\left(0\right)=3,k\left(2\right)=-5,k\left(-2\right)=3,k\left(\sqrt{2}\right)=1-2\sqrt{2},k\left(a+2\right)=-a^2-6a-5,k\left(-x\right)=-x^2+2x+3,\\ k\left(x^2\right)=-x^4-2x^2+3\end{array}$.

Step-by-step solution

Step 1. Evaluate k0,k2,k−2,k2 for the given function.

From the given function,

$k\left(x\right)=-x^2-2x+3....\left(1\right)$

Evaluate the value of function at $x=0,2,-2$gives,

role="math" localid="1644504817782" $$\begin{gathered} k\left(0\right)=-{\left(0\right)}^2-2\left(0\right)+3....\left(\text{Substitute}x=0\text{in}\left(1\right)\right) \\ =3 \\ k\left(2\right)=-{\left(2\right)}^2-2\left(2\right)+\mathrm{3....}\left(\text{Substitute}x=2\text{\hspace{0.17em}in}\left(1\right)\right) \\ =-5 \\ k\left(-2\right)=-{\left(-2\right)}^2-2\left(-2\right)+3....\left(\text{Substitute}x=-2\text{in}\left(1\right)\right) \\ =3 \\ k\left(\sqrt{2}\right)=-{\left(\sqrt{2}\right)}^2-2\left(\sqrt{2}\right)+3....\left(\text{Substitute}x=\sqrt{2}\text{in}\left(1\right)\right) \\ =1-2\sqrt{2} \end{gathered}$$

Step 2. Evaluate ka+2,k−x,kx2 for the given function.

Similarly, evaluate the value of function (1) at $x=a+2,-x,x^2$gives,

role="math" localid="1644505036706" $$\begin{gathered} k\left(a+2\right)=-{\left(a+2\right)}^2-2\left(a+2\right)+3....\left(\text{Substitute}x=a+2\text{in}\left(1\right)\right) \\ =-a^2-4a-4-2a-4+3 \\ =-a^2-6a-5 \\ k\left(-x\right)=-{\left(-x\right)}^2-2\left(-x\right)+\mathrm{3....}\left(\text{Substitute}x=-x\text{\hspace{0.17em}in}\left(1\right)\right) \\ =-x^2+2x+3 \\ k\left(x^2\right)=-{\left(x^2\right)}^2-2\left(x^2\right)+3....\left(\text{Substitute}x=x^2\text{in}\left(1\right)\right) \\ =-x^4-2x^2+3 \end{gathered}$$

Step 3. Description of steps.

From the obtained values,

$\begin{array}{l}k\left(0\right)=3,k\left(2\right)=-5,k\left(-2\right)=3,k\left(\sqrt{2}\right)=1-2\sqrt{2},k\left(a+2\right)=-a^2-6a-5,k\left(-x\right)=-x^2+2x+3,\\ k\left(x^2\right)=-x^4-2x^2+3\end{array}$.