(1) <em>f(x)</em> = (1 - <em>x</em>³) / (<em>x</em> - 1)
(a) The domain is the set of values that this function can take on. If <em>x</em> = 1, the denominator becomes 0 and the function is undefined. Any other value of <em>x</em> is okay, though, since for <em>x</em> ≠ 1, we have
<em>f(x)</em> = (1 - <em>x</em>³) / (<em>x</em> - 1) = - (1 - <em>x</em>³) / (1 - <em>x</em>) = -(<em>x</em>² + <em>x</em> + 1)
which is defined for all <em>x</em>. This also tells us that the plot of <em>f(x)</em> is a parabola with a hole at <em>x</em> = 1. So, the domain is the interval (-∞, 1) ∪ (1, ∞).
(b) The range is the set of values that the function actually does take on. Taking the simplified version of <em>f(x)</em>, we can complete the square to write
-(<em>x</em>² + <em>x</em> + 1) = -(<em>x</em>² + <em>x</em> + 1/4 - 1/4) - 1 = -(<em>x</em> + 1/2)² - 3/4
which is represented by a parabola that opens downward, with a maximum value of -3/4. So the range is the interval (-∞, -3/4).
(c) Judging by the plot of <em>f</em>, the limits at both negative and positive infinity are -∞.
(d) Same answer as part (a).
(2) <em>f(x)</em> = <em>x</em>³ - <em>x</em>
(a) The derivative of <em>f</em> at <em>x</em> = 3, and hence the slope of the tangent line to this point, is
(b) The tangent line at <em>x</em> = 3 has equation
<em>y</em> - <em>f </em>(3) = <em>f ' </em>(3) (<em>x</em> - 3)
<em>y</em> - 24 = 26 (<em>x</em> - 3)
<em>y</em> = 26 <em>x</em> - 54
We also want to find any other tangent lines parallel to this one, which requires finding all <em>x</em> for which <em>f '(x)</em> = 26. We could use the same limit definition as in part (a), but to save time, we exploit the power rule to get
<em>f</em> <em>'(x)</em> = 3<em> x</em>² - 1
Then solve for when this is equal to 26:
3<em> x</em>² - 1 = 26 ==> <em>x</em>² = 9 ==> <em>x</em> = ±3
The other tangent line occurs at <em>x</em> = -3, for which we have <em>f</em> (-3) = -24, and so the equation for the tangent is
<em>y</em> - <em>f</em> (-3) = 26 (<em>x</em> - (-3))
<em>y</em> + 24 = 26 (<em>x</em> + 3)
<em>y</em> = 26 <em>x</em> + 54