Question 2
Solve the following inequalities:
(a)\(|x-4|<3\)
(b)\(|2x+1|\geq5\)
(c)\(\left|\dfrac{x-1}{x+2}
ight|<1,\quad x
eq-2\)
(a)
(b)
Answer
\(x\leq-3\) or \(x\geq2\)
(c)
Square (both sides positive)
\((x-1)^2<(x+2)^2\Rightarrow x^2-2x+1
- frac{1}{2}\)Answer
\(x>- frac{1}{2},\;x
eq-2\)
Question 3
Sketch \(y=|x^2-2x-3|\), labelling all intercepts.
Base graph
\(y=x^2-2x-3=(x-3)(x+1)\); roots at \(x=-1,3\); vertex at \((1,-4)\)
Apply modulus
Reflect the portion below the \(x\)-axis upward; vertex becomes \((1,4)\)
Key points
\(x\)-ints at \((-1,0)\) and \((3,0)\); local max at \((1,4)\); \(y\)-int at \((0,3)\)
Question 4
Given \(f(x)=x^2-4\), sketch \(y=\dfrac{1}{f(x)}\), stating all asymptotes and intercepts.
VA where \(f(x)=0\)
\(x=\pm2\)
HA
\(y=0\) (degree of denominator exceeds numerator)
\(y\)-int
\(y= frac{1}{-4}=- frac{1}{4}\)
Local min/max
Where \(f(x)\) has vertex \((0,-4)\): reciprocal has local max \((0,- frac{1}{4})\)
Sketch features
VA \(x=\pm2\), HA \(y=0\), local max \((0,- frac{1}{4})\)
Question 5
Sketch \(y=|2x-1|\) and \(y=x+1\) on the same axes. Hence solve \(|2x-1|=x+1\).
Intersections
Case 1: \(2x-1=x+1\Rightarrow x=2\); Case 2: \(-(2x-1)=x+1\Rightarrow-2x+1=x+1\Rightarrow x=0\)
Check validity
Both valid: at \(x=2\): \(y=3\); at \(x=0\): \(y=1\)
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