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Функције - знак функције 2

Функције. Знак функције. Одређивање домена, нула и знака функције на сложенијим примерима.

Задаци

Текст задатака објашњених у видео лекцији:

пр.7)  $y={\log _3}{(1-2x)}$

пр.8)  $y={\log _{\frac{1}{2}}}\frac{{2x - 1}}{{2 - x}}$

пр.7)

\[\begin{gathered}
Df:1 - 2x > 0 \hfill \\
- 2x > - 1 \hfill \\
x < \frac{1}{2} \hfill \\
Df:x \in \left( { - \infty ;\frac{1}{2}} \right) \hfill \\
\end{gathered} \]

нуле:

\[\begin{gathered}
y = 0 \hfill \\
{\log _3}(1 - 2x) = 0 \hfill \\
1 - 2x = 1 \hfill \\
- 2x = 0 \hfill \\
x = 0 \in Df \hfill \\
\end{gathered} \]

знак:

429 png

 

\[\begin{gathered}
\begin{array}{*{20}{c}}
{\underline {y < 0} }&{}&{0 < 1 - 2x < 1} \\
{}&{}&{\begin{array}{*{20}{c}}
{1 - 2x > 0}&{}&{1 - 2x < 1} \\
{ - 2x > - 1}&{}&{ - 2x < 0} \\
{x < \frac{1}{2}}&{}&{x > 0}
\end{array}} \\
{}&{}&{x \in \left( {0;\frac{1}{2}} \right)}
\end{array} \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
{\underline {y > 0} }&{}&{1 - 2x > 1} \\
{}&{}&{ - 2x > 0} \\
{}&{}&{x < 0} \\
{}&{}&{x \in \left( { - \infty ;0} \right)}
\end{array} \hfill \\
\end{gathered} \]

8)

\[\begin{array}{*{20}{c}}
{Df:\frac{{2x - 1}}{{2 - x}} > 0}& \cap &{2 - x \ne 0} \\
{\begin{array}{*{20}{c}}
{2x - 1 = 0}&{}&{2 - x \ne 0} \\
{2x = 1}&{}&{x \ne 2} \\
{x = \frac{1}{2}}&{}&{}
\end{array}}&{}&{x \ne 2} \\
{}&{}&{}
\end{array}\]

430 png

\[Df:x \in \left( {\frac{1}{2};2} \right)\]

нуле:

\[\begin{gathered}
y = 0 \hfill \\
{\log _{\frac{1}{2}}}\frac{{2x - 1}}{{2 - x}} = 0 \hfill \\
\frac{{2x - 1}}{{2 - x}} = 1 \hfill \\
\frac{{2x - 1}}{{2 - x}} - 1 = 0 \hfill \\
\frac{{2x - 1 - \left( {2 - x} \right)}}{{2 - x}} = 0 \hfill \\
\frac{{2x - 1 - 2 + x}}{{2 - x}} = 0 \hfill \\
\frac{{3x - 3}}{{2 - x}} = 0 \hfill \\
3x - 3 = 0 \hfill \\
x = 1 \in Df \hfill \\
\end{gathered} \]

знак:

431 png

\[\begin{array}{*{20}{c}}
{\underline {y > 0} }&{}&{0 < \frac{{2x - 1}}{{2 - x}} < 1} \\
{}&{}&{\begin{array}{*{20}{c}}
\begin{gathered}
\frac{{2x - 1}}{{2 - x}} > 0 \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
{2x - 1 = 0}&{}&{2 - x \ne 0} \\
{2x = 1}&{}&{x \ne 2} \\
{x = \frac{1}{2}}&{}&{}
\end{array} \hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
\cap \hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
\frac{{2x - 1}}{{2 - x}} < 1 \hfill \\
\frac{{2x - 1}}{{2 - x}} - 1 < 0 \hfill \\
\frac{{2x - 1 - 2 + x}}{{2 - x}} < 0 \hfill \\
\frac{{3x - 3}}{{2 - x}} < 0 \hfill \\
\begin{array}{*{20}{c}}
{3x - 3 = 0}&{}&{2 - x \ne 0} \\
{x = 1}&{}&{x \ne 2}
\end{array} \hfill \\
\end{gathered}
\end{array}}
\end{array}\]

432 png

433 png

\[\begin{array}{*{20}{c}}
\begin{gathered}
\underline {y < 0} \hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &{}&\begin{gathered}
\frac{{2x - 1}}{{2 - x}} > 1 \hfill \\
\frac{{2x - 1}}{{2 - x}} - 1 > 0 \hfill \\
\frac{{2x - 1 - 2 + x}}{{2 - x}} > 0 \hfill \\
\frac{{3x - 3}}{{2 - x}} > 0 \hfill \\
x \in \left( {1;2} \right) \hfill \\
\end{gathered}
\end{array}\]