Mister Exam
Graphing y = log(x+10)/log(15)-|x-25|*((x+7)/x)^(13/2)
The solution
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13/2
log(x + 10) /x + 7\
f(x) = ----------- - |x - 25|*|-----|
log(15) \ x /
$$f{\left(x \right)} = - \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}$$
f = -((x + 7)/x)^(13/2)*|x - 25| + log(x + 10)/log(15)
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -8.99482842928553$$
$$x_{2} = -8.99482842928573$$
$$x_{3} = -8.99482842928543$$
$$x_{4} = -8.99482842928432$$
$$x_{5} = -8.9948284292855$$
$$x_{6} = -8.99482842928557$$
$$x_{7} = 24.7405918283319$$
$$x_{8} = 25.2684484737108$$
$$x_{9} = -8.99482842928554$$
so we need to solve the equation:
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -8.99482842928553$$
$$x_{2} = -8.99482842928573$$
$$x_{3} = -8.99482842928543$$
$$x_{4} = -8.99482842928432$$
$$x_{5} = -8.9948284292855$$
$$x_{6} = -8.99482842928557$$
$$x_{7} = 24.7405918283319$$
$$x_{8} = 25.2684484737108$$
$$x_{9} = -8.99482842928554$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(x + 10)/log(15) - |x - 25|*((x + 7)/x)^(13/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
$$\lim_{x \to -\infty}\left(\frac{- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = - \left(- \frac{7 - x}{x}\right)^{\frac{13}{2}} \left|{x + 25}\right| + \frac{\log{\left(10 - x \right)}}{\log{\left(15 \right)}}$$
- No
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = \left(- \frac{7 - x}{x}\right)^{\frac{13}{2}} \left|{x + 25}\right| - \frac{\log{\left(10 - x \right)}}{\log{\left(15 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = - \left(- \frac{7 - x}{x}\right)^{\frac{13}{2}} \left|{x + 25}\right| + \frac{\log{\left(10 - x \right)}}{\log{\left(15 \right)}}$$
- No
$$- \left(\frac{x + 7}{x}\right)^{\frac{13}{2}} \left|{x - 25}\right| + \frac{\log{\left(x + 10 \right)}}{\log{\left(15 \right)}} = \left(- \frac{7 - x}{x}\right)^{\frac{13}{2}} \left|{x + 25}\right| - \frac{\log{\left(10 - x \right)}}{\log{\left(15 \right)}}$$
- No
so, the function
not is
neither even, nor odd