22. For each of the following functions, state:
i) domain iv) equations of any asymptotes
ii) x-intercepts v) location of any removable discontinuities
iii) y-intercept vi) analyze the behaviour around the asymptotes
a)
2 1
2
x
x
f x
b)
1
1
2
x
x
f x
c)
2 7 6
3 5
2
x x
x
f x d)
2 8
1
2
2
x
a)
i) The domain of \(f(x)\) is all real numbers except \(x=0\) and \(x=1\).
ii) The x-intercept occurs when \(f(x)=0\), so we solve \(2x^2-x+1=0\). However, there are no real solutions for this equation, so there are no x-intercepts.
iii) The y-intercept occurs when \(x=0\), so we have \(f(0) = 1\).
iv) There are no horizontal asymptotes because as \(x\) approaches positive or negative infinity, \(f(x)\) does not approach a constant value.
v) There are no removable discontinuities.
vi) Around the vertical asymptote \(x=1\), the function approaches negative infinity as \(x\) approaches 1 from the left (\(x<1\)), and approaches positive infinity as \(x\) approaches 1 from the right (\(x>1\)).
b)
i) The domain of \(f(x)\) is all real numbers except \(x=0\) and \(x=1\).
ii) The x-intercept occurs when \(f(x)=0\), so we solve \(x^2-2x+1=0\). Factoring, we get \((x-1)^2=0\), so the x-intercept is \(x=1\).
iii) The y-intercept occurs when \(x=0\), so we have \(f(0) = 1\).
iv) There are no horizontal asymptotes because as \(x\) approaches positive or negative infinity, \(f(x)\) does not approach a constant value.
v) There are no removable discontinuities.
vi) Around the vertical asymptote \(x=1\), the function approaches negative infinity as \(x\) approaches 1 from the left (\(x<1\)), and approaches positive infinity as \(x\) approaches 1 from the right (\(x>1\)).
c)
i) The domain of \(f(x)\) is all real numbers.
ii) The x-intercepts occur when \(f(x)=0\), so we solve \(2x^3-7x^2+6x=0\). Factoring, we get \(x(2x-1)(x-6)=0\), so the x-intercepts are \(x=0\), \(x=\frac{1}{2}\), and \(x=6\).
iii) The y-intercept occurs when \(x=0\), so we have \(f(0) = 0\).
iv) There are no horizontal asymptotes because as \(x\) approaches positive or negative infinity, \(f(x)\) does not approach a constant value.
v) There are no removable discontinuities.
vi) There are no vertical asymptotes.
d)
i) The domain of \(f(x)\) is all real numbers except \(x=2\) and \(x=-8\).
ii) The x-intercepts occur when \(f(x)=0\), so we solve \((x+2)(x-8)=0\), giving us x= -2 and x=8.
iii) The y-intercept occurs when \(x=0\), so we have \(f(0) = 4\).
iv) There are no horizontal asymptotes because as \(x\) approaches positive or negative infinity, \(f(x)\) does not approach a constant value.
v) There are no removable discontinuities.
vi) There are no vertical asymptotes.