Chapter 2.2, Problem 23AYU

In problems $25-30$, answer the questions about each function.

$f(x)=3x^2+x-2$

Is the point $(1,2)$ on the graph of $f$?

If $x=-2$, what is $f(x)$? What point is on the graph of $f?$

If $f(x)=-2,$ what is x? What point(s) are on the graph of $f?$

What is the domain of $f?$

List the $x-$ intercepts, if any, of the graph of $f?$

List the $y-$ intercept, if there is one, of the graph of $f?$

To determine

The point $\left(1,2\right)$ on the graph of $f$ if $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

Yes, the point $\left(1,2\right)$ is on the graph of $f\left(x\right)=3x^2+x-2$.

Explanation of Solution

Given Information:

$f\left(x\right)=3x^2+x-2$, and the point $\left(1,2\right)$.

Explanation:

If $\left(x,y\right)$ is a point on the graph of a function $f$, then $y$ is the value of $f$ at $x$, that is, $y=f(x)$.

Substitute $x=1$ in $f\left(x\right)=3x^2+x-2$.

$f\left(1\right)=3{\left(1\right)}^2+1-2$,

$f\left(1\right)=3+1-2=4-2$,

$\Rightarrow f\left(1\right)=2$.

Thus, the point $\left(1,2\right)$ is on the graph of $f\left(x\right)=3x^2+x-2$.

To determine

$f\left(x\right)$ if $x=-2$. Find the point on the graph of $f$ if $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

$f\left(x\right)=8$ when $x=-2$. The point $\left(-2,8\right)$ is on the graph of $f$.

Explanation of Solution

Given Information:

$f\left(x\right)=3x^2+x-2$ if $x=-2$.

Explanation:

If $x=-2$ then,

$f\left(-2\right)=3{\left(-2\right)}^2+\left(-2\right)-2$,

$f\left(-2\right)=12-2-2=12-4$,

$\Rightarrow f\left(-2\right)=8$.

Thus, $f\left(x\right)=8$ if $x=-2$.

The point $\left(-2,8\right)$ is on the graph $f$.

To determine

$x$ if $f\left(x\right)=-2$. Find the points on the graph of $f$ if $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

If $f\left(x\right)=-2$, then $x=0,-\frac{1}{3}$, and the points $\left(0,-2\right),\left(-\frac{1}{3},-2\right)$ are on the graph of $f\left(x\right)=3x^2+x-2$.

Explanation of Solution

Given Information:

$f\left(x\right)=3x^2+x-2$, and $f\left(x\right)=-2$.

Explanation:

Substitute $f\left(x\right)=-2$ in $f\left(x\right)=3x^2+x-2$.

$f\left(x\right)=3x^2+x-2$,

$-2=3x^2+x-2$,

$-2+2=3x^2+x$,

$3x^2+x=0$.

Factor out $x$ from the left side of the above equation,

$x\left(3x+1\right)=0$,

$\Rightarrow x=0$ or $3x+1=0$,

$3x+1=0\Rightarrow 3x=-1$.

$x=0$ or $x=-\frac{1}{3}$.

If $f\left(x\right)=-2$, then $x=0$ or $x=-\frac{1}{3}$. The points $\left(0,-2\right)$ and $\left(-\frac{1}{3},-2\right)$ are on the graph of $f\left(x\right)=3x^2+x-2$.

Thus, $f\left(x\right)=-2$, then $x=0$ or $x=-\frac{1}{3}$. The points $\left(0,-2\right)$ and $\left(-\frac{1}{3},-2\right)$ are on the graph of $f\left(x\right)=3x^2+x-2$.

To determine

The domain of $f$ if $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

The domain of $f\left(x\right)=3x^2+x-2$ is $\left(-\infty ,\infty \right)$.

Explanation of Solution

Given Information:

$f\left(x\right)=3x^2+x-2$.

Explanation:

$f\left(x\right)=3x^2+x-2$ is the quadratic function, so the graph of this function is a parabola.

The domain of the function is the set of all values of $x$ for which the function is defined.

Since it is a quadratic function, it is defined for all values of $x$.

Thus, domain of the function $f\left(x\right)=3x^2+x-2$ is all real numbers.

In interval notation, the domain would be $\left(-\infty ,\infty \right)$.

To determine

The $x$-intercepts, if any, of the graph of $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

The $x$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$ are $\left(-1,0\right),\left(\frac{2}{3},0\right)$.

Explanation of Solution

Given Information:

The function is $f\left(x\right)=3x^2+x-2$.

Explanation:

The $x$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$ are the real solutions of the equation $f\left(x\right)=0$ that are in the domain of $f$.

$f\left(x\right)=3x^2+x-2=0$.

$3x^2+x-2=0$.

Find the factors of this quadratic equation.

$3x^2+3x-2x-2=0$,

$3x\left(x+1\right)-2\left(x+1\right)=0$,

$\left(x+1\right)\left(3x-2\right)=0$,

$x+1=0$ or $3x-2=0$.

$\Rightarrow x=-1$ or $x=\frac{2}{3}$.

Thus, the $x$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$ are $\left(-1,0\right),\left(\frac{2}{3},0\right)$.

To determine

The $y$-intercepts, if any, of the graph of $f\left(x\right)=3x^2+x-2$.

Answer to Problem 23AYU

Solution:

The $x$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$ are $\left(0,-2\right)$.

Explanation of Solution

Given Information:

The function is $f\left(x\right)=3x^2+x-2$.

Explanation:

To find the $y$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$, substitute $x=0$ in $f\left(x\right)$.

$f\left(x\right)=3x^2+x-2$.

Substitute $x=0$ in above function

$f\left(0\right)=3{\left(0\right)}^2+0-2=-2$

$\Rightarrow f\left(0\right)=-2$

Thus, the $y$-intercepts of the graph of $f\left(x\right)=3x^2+x-2$ are $\left(0,-2\right)$.