Answer
Evaluate $k(x) = 2x^3 - 3x^2$:
$k(0) = 0$
$k(3) = 27$
$k(-3) =-81$
$k(\frac{1}{2}) = -\frac{1}{2}$
$k(\frac{a}{2}) =\frac{a^3-3a^2}{4}$
$k(-x) = -2x^3 - 3x^2$
$k(x^3) =2x^9 - 3x^6$
Work Step by Step
For x = 0
$k(0) = 2(0)^3 - 3(0)^2$ ... evaluate exponents
$ = 2(0) - 3(0)$ ... multiply
$=0 - 0 = 0$
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For x = 3
$k(3) = 2(3)^3 - 3(3)^2$ ... evaluate exponents
$ = 2(27) - 3(9)$ ... multiply
$=54 - 27 = 27$
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For x = -3
$k(-3) = 2(-3)^3 - 3(-3)^2$ ... evaluate exponents
$ = 2(-27) - 3(9)$ ... multiply
$=-54 - 27 = -81$
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For x = $\frac{1}{2}$
$k(\frac{1}{2}) = 2(\frac{1}{2})^3 - 3(\frac{1}{2})^2$ ... distribute exponents
$ = 2(\frac{1^3}{2^3}) - 3(\frac{1^2}{2^2})$ ... evaluate exponents
$=2(\frac{1}{8}) - 3(\frac{1}{4})$ ... multiply
$=\frac{2}{8} - \frac{3}{4}$ ... simplify for common denominator
$=\frac {1}{4} - \frac{3}{4}$ ... subtract
$=-\frac{2}{4}= - \frac{1}{2}$
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For x = $\frac{a}{2}$
$k(\frac{a}{2}) = 2(\frac{a}{2})^3 - 3(\frac{a}{2})^2$ ... distribute exponents
$ = 2(\frac{a^3}{2^3}) - 3(\frac{a^2}{2^2})$ ... evaluate exponents
$=2(\frac{a^3}{8}) - 3(\frac{a^2}{4})$ ... multiply
$=\frac{2a^3}{8} - \frac{3a^2}{4}$ ... simplify for common denominator
$=\frac {a^3}{4} - \frac{3a^2}{4}$ ... subtract
$=\frac{a^3-3a^2}{4}$
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For x = $-x$
$k(-x) = 2(-x)^3 - 3(-x)^2$ ... evaluate exponents
$ = 2(-x^3) - 3(x^2)$ ... multiply
$=-2x^3 - 3x^2$
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For x = $x^3$
$k(x^3) = 2(x^3)^3 - 3(x^3)^2$ ... evaluate exponents
$ = 2(x^9) - 3(x^6)$ ... multiply
$=2x^9 - 3x^6$
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