Questions: Four functions are given below. Either the function is defined explicitly, or the entire graph of the function is shown. For each, decide whether it is an even function, an odd function, or neither. The function r The function s Even g(x)=7x^4-4x^3 Odd Neither Even h(x)=6x^3 Odd Neither

Four functions are given below. Either the function is defined explicitly, or the entire graph of the function is shown.

For each, decide whether it is an even function, an odd function, or neither.
The function r The function s
Even g(x)=7x^4-4x^3 Odd
Neither Even
h(x)=6x^3 Odd
Neither

Transcript text: Four functions are given below. Either the function is defined explicitly, or the entire graph of the function is shown. For each, decide whether it is an even function, an odd function, or neither. \begin{tabular}{|c|c|} \hline The function $r$ & The function $s$ \\ \hline Even $g(x)=7 x^{4}-4 x^{3}$ Odd Neither & \begin{tabular}{l} Even \\ $h(x)=6 x^{3}$ Odd Neither \end{tabular} \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Analyze the function $ g(x) = 7x^4 - 4x^3 $

To determine if $ g(x) $ is even, check if $ g(-x) = g(x) $: \[ g(-x) = 7(-x)^4 - 4(-x)^3 = 7x^4 + 4x^3 \] Since $ g(-x) \neq g(x) $, $ g(x) $ is not even.
To determine if $ g(x) $ is odd, check if $ g(-x) = -g(x) $: \[ g(-x) = 7x^4 + 4x^3 \neq -7x^4 + 4x^3 = -g(x) \] Since $ g(-x) \neq -g(x) $, $ g(x) $ is not odd.
Conclusion: $ g(x) $ is neither even nor odd.

Step 2: Analyze the function $ h(x) = 6x^3 $

To determine if $ h(x) $ is even, check if $ h(-x) = h(x) $: \[ h(-x) = 6(-x)^3 = -6x^3 \] Since $ h(-x) \neq h(x) $, $ h(x) $ is not even.
To determine if $ h(x) $ is odd, check if $ h(-x) = -h(x) $: \[ h(-x) = -6x^3 = -h(x) \] Since $ h(-x) = -h(x) $, $ h(x) $ is odd.
Conclusion: $ h(x) $ is odd.

Step 3: Analyze the graph of function $ r $