Question

Precalc question
(please show work)

Answer 1

(a) Even

(b) Neither

Explanation:

Even and odd functions are special types of functions.

Even function

• f(-x) = f(x) for all values of x.
• Symmetric about the y-axis.
• Example even function: y = x²

Odd function

• f(-x) = -f(x) for any value of x.
• Symmetric about the origin.
• Example odd function: y = x³

To determine if a function is even, odd, or neither, we need to consider how the function behaves with respect to symmetry by replacing x with -x in the function and observing if the function remains unchanged (even function), becomes negative (odd function), or does something else (neither).

Part (a)

Given function:

`r(x)=√(7-x^2)`

Replace the x with -x:

`r(-x)=√(7-(-x)^2)`

`r(-x)=√(7-x^2)`

`r(-x)=r(x)`

As (-x)² = x² for all values of x, then r(-x) = r(x) for all values of x.

Therefore, function r(x) is even.

Note: The graph of function r(x) is symmetric about the y-axis.

Part (b)

Given function:

`h(x)=8x^3-5x+2`

Replace the x with -x:

`h(-x)=8(-x)^3-5(-x)+2`

`h(-x)=-8x^3+5x+2`

As h(-x) ≠ h(x) and h(-x) ≠ -h(x) for all values of x, function h(x) is neither an even function nor an odd function.

Note: The graph of function h(x) is not symmetric about origin.