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Section 4.5 Exercises

True or False.  Determine whether each pair of expressions is equivalent.

1.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \sqrt{{x}^{2}+4} &\leftrightarrow &  x+2 \end{array}[/latex]

2.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ {\left(3{e}^{x}\right)}^{2} &\leftrightarrow &  9{e}^{{x}^{2}} \end{array}[/latex]

3.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \frac{{x}^{2}+5}{{x}^{2}} &\leftrightarrow & 5  \end{array}[/latex]

4.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \frac{{x}^{2}+5}{{x}^{2}} &\leftrightarrow & 1+\frac{5}{{x}^{2}}  \end{array}[/latex]

5.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \sqrt{4x+8} &\leftrightarrow & 4\sqrt{x+2}  \end{array}[/latex]

6. [latex]\begin{array}[b]{ccc} & \text{?} & \\ {e}^{ln\left(x\right)} &\leftrightarrow & x  \end{array}[/latex]

7. [latex]\begin{array}[b]{ccc} & \text{?} & \\ {sin}\left(3x\right) &\leftrightarrow & 3{sin}\left(x\right)  \end{array}[/latex]

8.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \sqrt{9{x}^{2}+40} &\leftrightarrow &  3\sqrt{{x}^{2}+5} \end{array}[/latex]

9.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \log\left(5x\right) &\leftrightarrow & 5\log\left(x\right)  \end{array}[/latex]

10.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \frac{{h}^{2}+2h}{h} &\leftrightarrow & {h}^{2}+2  \end{array}[/latex]

11.  [latex]\begin{array}[b]{ccc} & \text{?} & \\ \frac{1}{x-2} &\leftrightarrow & \frac{1}{x}-\frac{1}{2}  \end{array}[/latex]

Answers to Section 4.5 Problems

1.  False.  [latex]{x}^{2}[/latex] and [latex]4[/latex] are added.  They cannot be evaluated separately.

2.  False.  [latex]{\left(3{e}^{x}\right)}^{2}=9{e}^{2x}[/latex]

3.  False.  [latex]{x}^{2}[/latex] in the numerator is added to 5.  It cannot be cancelled before we add.  Only common factors in the numerator and denominator can be cancelled – NOT common terms.

4.  True

5.  False.  The 4 can be factored from the terms inside the radical.  However, it can’t leave the radical without taking the square root.  The equivalent expression here would be [latex]2\sqrt{x+2}[/latex].

6.  True.  In fact, anytime a number [latex]b[/latex] is raised to the log of the same base, the result is always the argument of the log.  That is, [latex]{b}^{\log_{b}\left(x\right)}=x[/latex] always.

7.  False.  [latex]3x[/latex] represents [latex]3[/latex] times an angle of measure [latex]x[/latex] degrees/radians around the unit circle.  [latex]3{sin}\left(x\right)[/latex] represents [latex]3[/latex] times the [latex]y-[/latex]coordinate of angle [latex]x[/latex].

8.  True.

9.  False.  [latex]\log\left(3x\right)=y[/latex] means [latex]3x={10}^{y}[/latex].  [latex]3\log\left(x\right)=y[/latex] means [latex]x={10}^{\frac{y}{3}}[/latex].

10.  False.  [latex]\frac{{h}^{2}+2h}{h}=\frac{\cancel{h}\left(h+2\right)}{\cancel{h}}=h+2[/latex]

11.  False.  [latex]\frac{1}{x}-\frac{1}{2}=\frac{2}{2x}-\frac{x}{2x}=\frac{2-x}{2x}[/latex]