Chapter 5, Problem 2CT

(a)

To determine

The function $y=4x^2+3$ is one-to-one or not.

Answer to Problem 2CT

Solution:

The function $y=4x^2+3$ is not a one-to-one function.

Explanation of Solution

Given information:

The functions is $y=4x^2+3$.

Explanation:

$f$ is said to be one-to-one if every horizontal line intersects the graph of a function $f$ in at most one point.

Graph the function $y=4x^2+3$ using graphing calculator as below:

Step $1$: Press “ $\text{Y=}$” button, and type the equation as $4x^2+3$.

Step $2$: Press “ $\text{Y=}$” button, and type $Y=5$ for plotting the horizontal line on the graph of $y=4x^2+3$.

Step $3$: Press “GRAPH” button to see the graph.

Thus, the graph is:

The horizontal line $Y=5$ intersects the graph of $f$ twice, at $\left(-0.707,5\right)$ and at $\left(0.707,5\right)$.

Therefore, the function $y=4x^2+3$ is not one-to-one.

(b)

To determine

The function $y=\sqrt{x+3}-5$ is one-to-one or not.

Answer to Problem 2CT

Solution:

The function $y=\sqrt{x+3}-5$ is a one-to-one function.

Explanation of Solution

Given information:

The functions is $y=\sqrt{x+3}-5$.

Explanation:

$f$ is said to be one-to-one if every horizontal line intersects the graph of a function $f$ in at most one point.

Graph the function $y=\sqrt{x+3}-5$ using calculator as below:

Step $1$: Press “ $\text{Y=}$” button, and type the equation as $\sqrt{x+3}-5$.

Step $2$: Press “ $\text{Y=}$” button, and type $Y=-3$ for plotting the horizontal line on the graph of $y=\sqrt{x+3}-5$.

Step $3$: Press “GRAPH” button to see the graph.

Thus, the graph is:

The horizontal line $Y=-3$ intersects the graph of $f$ once, at $\left(1,-3\right)$.

Therefore, the function $y=\sqrt{x+3}-5$ is one-to-one.