Mathski — Mixed Functions SL II

Mixed Functions SL II

IB Mathematics AA · SL · Shadow Worksheet

Practice

Name

Question 1

Find the exact solutions to \(2e^{2x}-5e^x+2=0\).

Substitution \(u=e^x\)

\(2u^2-5u+2=0\Rightarrow(2u-1)(u-2)=0\Rightarrow u= frac{1}{2}\) or \(u=2\)

Answer

\(x=\ln frac{1}{2}=-\ln2\) or \(x=\ln2\)

Question 2

Find the range of \(g(x)=3-2\sin(x)\).

Range of \(\sin x\)

\(-1\leq\sin x\leq1\Rightarrow-2\leq-2\sin x\leq2\)

Answer

\(1\leq g(x)\leq5\)

Question 3

Let \(f(x)=\ln(x+a)\). Given \(f^{-1}(0)=2\), find \(a\). Hence find the domain and range of \(f\).

\(f^{-1}(0)=2\) means \(f(2)=0\)

\(\ln(2+a)=0\Rightarrow2+a=1\Rightarrow a=-1\)

Answer

\(a=-1\); \(f(x)=\ln(x-1)\); Domain: \(x>1\); Range: \(\mathbb{R}\)

Question 4

The graph of \(y=f(x)\) is shown with key points \((-2,3),(0,1),(1,-2)\). Sketch \(y=-f(x+1)\) labelling the images of these three points.

Apply transformations in order

Translate 1 left: \((-3,3),(−1,1),(0,-2)\). Then reflect in \(x\)-axis: \((-3,-3),(-1,-1),(0,2)\)

Answer

Images: \((-3,-3),(-1,-1),(0,2)\)

Question 5

Solve \(\log_2(x+3)+\log_2(x-1)=\log_2(x+9)\).

Combine left side

\(\log_2[(x+3)(x-1)]=\log_2(x+9)\Rightarrow x^2+2x-3=x+9\Rightarrow x^2+x-12=0\Rightarrow(x+4)(x-3)=0\)

Check validity

\(x=-4\): \(\log_2(-1)\) undefined. Reject.

Answer

\(x=3\)

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