Determine whether the function is one-to-one.

(a)$y=4x^2+3$

(b)$y=\sqrt{x+3}-5$

We want to Determine whether the function is one-to-one.

(a)$4x^2+3$

(b)$\sqrt{x+3}-5$

To check if a function is one to one we must first draw a graph of the function. (The function is one to one if each horizontal line intersects a graph of the function f in at most one point.)

Let's draw a quadratic function $f\left(x\right)=4x^2+3$

• Since the function f is created so that the graph of the function $4x^2$shifts 3 upwards along the y-axis, then the vertex of the parabola in V(0,3).
• Take in the table ea few x's that are around x=0 (the graph of the quadratic function is symmetric with respect to the $x=x_{VERTEX}$)

The graph is given by:

If we draw a horizontal line y=10 we see how the line intersects the function f at two points. (By the horizontal test we see that the function is not one to one because it mapped two different domain elements into the same codomain element).

Or, we look at the table we see that for example $x=-1andx=1$the functional values are equal, i.e. $f(-1)=f\left(1\right)=7$.

From which we conclude by definition that a function is not one to one.

To check if a function is one to one we must first draw a graph of the function. (The function is one to one if each horizontal line intersects a graph of the function f in at most one point.)

Let's draw a root function:

• To draw the function $f\left(x\right)=\sqrt{x+3}-5$is to make a table and x take such numbers that the expression under the root is non-negative (i.e. $x+3\ge 0$) and that x+3 is a square of some number for easier calculation $(x+3=0,x+3=1,x+3=4.x+3=9,...)$

The graph is given by

If we draw a horizontal line e.g. y=-4 we see how the line intersects the function f in one point (by the horizontal test we see that the function is one to one )

Also we could conclude that the function is one to one because the square root is a strictly increasing function.