Mixed Functions SL I
IB Mathematics AA · SL · Shadow Worksheet
Practice
Question 1
The function \(f(x)=\dfrac{a}{x+b}+c\) has a vertical asymptote at \(x=2\), horizontal asymptote at \(y=-1\), and passes through \((3,4)\). Find \(a,b,c\).
Asymptotes
VA: \(b=-2\); HA: \(c=-1\)
Through \((3,4)\)
\(\dfrac{a}{1}-1=4\Rightarrow a=5\)
Answer
\(a=5,\;b=-2,\;c=-1\)
Question 2
Let \(f(x)=\sqrt{2x+6}\).
(a)State the domain and range of \(f\).
(b)Find \(f^{-1}(x)\) and state its domain.
(c)Solve \(f(x)=f^{-1}(x)\).
(a)
Answer
Domain: \(x\geq-3\); Range: \(f(x)\geq0\)
(b)
Inverse
\(y=\sqrt{2x+6}\Rightarrow y^2=2x+6\Rightarrow x=\dfrac{y^2-6}{2}\)
Answer
\(f^{-1}(x)=\dfrac{x^2-6}{2},\;x\geq0\)
(c)
On \(y=x\)
\(\sqrt{2x+6}=x\Rightarrow 2x+6=x^2\Rightarrow x^2-2x-6=0\Rightarrow x=1\pm\sqrt{7}\); reject negative
Question 3
Given \(h(x)=f(g(x))\) where \(f(x)=2^x\) and \(g(x)=3x-1\), find \(h(x)\) and solve \(h(x)=8\).
Solve
\(2^{3x-1}=8=2^3\Rightarrow3x-1=3\Rightarrow x= frac{4}{3}\)
Answer
\(h(x)=2^{3x-1};\;x= frac{4}{3}\)
Question 4
Functions \(p\) and \(q\) are defined by \(p(x)=x^2-1\) and \(q(x)=2x+3\). Solve \(p(q(x))=q(p(x))\).
Compute each
\(p(q(x))=(2x+3)^2-1=4x^2+12x+8\); \(q(p(x))=2(x^2-1)+3=2x^2+1\)
Set equal
\(4x^2+12x+8=2x^2+1\Rightarrow2x^2+12x+7=0\Rightarrow x=\dfrac{-12\pm\sqrt{144-56}}{4}=\dfrac{-12\pm\sqrt{88}}{4}\)
Answer
\(x=\dfrac{-6\pm\sqrt{22}}{2}\)
Question 5
A function is defined by \(f(x)=egin{cases}3x+2 & x<1 \ x^2+1 & x\geq1\end{cases}\). Find \(f(-2),f(1),f(3)\). Is \(f\) continuous at \(x=1\)?
Evaluate
\(f(-2)=3(-2)+2=-4\); \(f(1)=1+1=2\); \(f(3)=10\)
Continuity at \(x=1\)
Left limit: \(3(1)+2=5\); Right limit/value: \(2\). Not equal.
Answer
\(f(-2)=-4,\;f(1)=2,\;f(3)=10\). Not continuous at \(x=1\).
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