Graphing y = sqrtx-4+1

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f(x) = \/ x  - 4 + 1

f{\left(x \right)} = \left(\sqrt{x} - 4\right) + 1

f = sqrt(x) - 4 + 1

The graph of the function

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(\sqrt{x} - 4\right) + 1 = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 9

Numerical solution

x_{1} = 9

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x) - 4 + 1.

\left(-4 + \sqrt{0}\right) + 1

The result:

f{\left(0 \right)} = -3

The point:

(0, -3)

Extrema of the function

In order to find the extrema, we need to solve the equation

\frac{d}{d x} f{\left(x \right)} = 0

(the derivative equals zero),
and the roots of this equation are the extrema of this function:

\frac{d}{d x} f{\left(x \right)} =

\frac{1}{2 \sqrt{x}} = 0

Solve this equation
Solutions are not found,
function may have no extrema

Inflection points

Let's find the inflection points, we'll need to solve the equation for this

\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0

(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:

\frac{d^{2}}{d x^{2}} f{\left(x \right)} =

- \frac{1}{4 x^{\frac{3}{2}}} = 0

Solve this equation
Solutions are not found,
maybe, the function has no inflections

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(\sqrt{x} - 4\right) + 1\right) = \infty i

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\left(\sqrt{x} - 4\right) + 1\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 sqrt(x) - 4 + 1, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left(\sqrt{x} - 4\right) + 1}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{x \to \infty}\left(\frac{\left(\sqrt{x} - 4\right) + 1}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left

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(\sqrt{x} - 4\right) + 1 = \sqrt{- x} - 3

- No

\left(\sqrt{x} - 4\right) + 1 = 3 - \sqrt{- x}

- No
so, the function
not is
neither even, nor odd