Problem 31 - Precalculus Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. \(h(x)=x^{4}-x^{2}+1\) a. \(h(2)\) b. \(h(-1)\) c. \(h(-x)\) d. \(h(3 a)\)

Step-by-Step Solution

Answer

The results of evaluating \(h\) at the given values are: a. \(h(2) = 13\), b. \(h(-1) = 1\), c. \(h(-x) = x^{4} - x^{2} + 1\), and d. \(h(3a) = 81a^{4} - 9a^{2} + 1\)

Step 1: Evaluate \(h(2)\)

Substitute \(x = 2\) into the function \(h(x)\):\(h(2) = (2)^{4} - (2)^{2} + 1 = 16 - 4 + 1 = 13\).

Step 2: Evaluate \(h(-1)\)

Substitute \(x = -1\) into the function \(h(x)\): \(h(-1) = (-1)^{4} - (-1)^{2} + 1 = 1 - 1 + 1 = 1\).

Step 3: Evaluate \(h(-x)\)

Substitute \(-x\) for \(x\) in the function \(h(x)\): \(h(-x) = (-x)^{4} - (-x)^{2} + 1 = x^{4} - x^{2} + 1\). This result is same to \(h(x)\), meaning that the function is an even function.

Step 4: Evaluate \(h(3a)\)

Substitute \(3a\) for \(x\) in the function \(h(x)\): \(h(3a) = (3a)^{4} - (3a)^{2} + 1 = 81a^{4} - 9a^{2} + 1\).