Questions: Evaluate the function h(x) = x^4 + 5x^2 + 3 at the given values of the independent variable and simplify. a. h(3) b. h(-1) c. h(-x) d. h(3a)

Evaluate the function h(x) = x^4 + 5x^2 + 3 at the given values of the independent variable and simplify. a. h(3) b. h(-1) c. h(-x) d. h(3a)
Transcript text: Evaluate the function $h(x)=x^{4}+5 x^{2}+3$ at the given values of the independent variable and simplify. a. $\mathrm{h}(3)$ b. $h(-1)$ c. $\mathrm{h}(-\mathrm{x})$ d. $h(3 a)$
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Solution

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Solution Steps

To evaluate the function \( h(x) = x^4 + 5x^2 + 3 \) at the given values, we will substitute the specified values into the function and compute the results.

a. Substitute \( x = 3 \) into \( h(x) \). b. Substitute \( x = -1 \) into \( h(x) \). c. Substitute \( x = -x \) into \( h(x) \).

Step 1: Evaluate \( h(3) \)

To evaluate \( h(3) \), we substitute \( x = 3 \) into the function: \[ h(3) = 3^4 + 5 \cdot 3^2 + 3 = 81 + 45 + 3 = 129 \]

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

Next, we evaluate \( h(-1) \) by substituting \( x = -1 \): \[ h(-1) = (-1)^4 + 5 \cdot (-1)^2 + 3 = 1 + 5 + 3 = 9 \]

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

For \( h(-x) \), we substitute \( x = -x \) into the function: \[ h(-x) = (-x)^4 + 5 \cdot (-x)^2 + 3 = x^4 + 5x^2 + 3 \]

Final Answer

The results for each evaluation are:

  • \( h(3) = 129 \)
  • \( h(-1) = 9 \)
  • \( h(-x) = x^4 + 5x^2 + 3 \)

Thus, the final answers are: \[ \boxed{h(3) = 129} \] \[ \boxed{h(-1) = 9} \] \[ \boxed{h(-x) = x^4 + 5x^2 + 3} \]

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