Question

Which of the following describes a true relationship between the functions $f\left( x \right) = {\left( {x - 3} \right)^2} + 2$ and $g\left( x \right) = \dfrac{1}{2}x + 1$ graphed below in the standard $\left( {x,y} \right)$ coordinate plane?

A) $f\left( x \right) = g\left( x \right)$ for exactly 2 values of x
B) $f\left( x \right) = g\left( x \right)$ for exactly 1 values of x
C) $f\left( x \right) < g\left( x \right)$ for all x
D) $f\left( x \right) > g\left( x \right)$ for all x
E) $f\left( x \right)$ is the inverse of $g\left( x \right)$

Answer 1

We can check the graph of the given function. The number of points that the 2 graphs will give the number of values the function will be equal. Then we can check whether a function is greater than another by checking whether the graph of one function is always above the other. Then we can check weather one function is the inverse of the other by checking whether the graphs are symmetric with respect to the line $x = y$

Complete step by step solution:
We have the 2 functions as functions $f\left( x \right) = {\left( {x - 3} \right)^2} + 2$ and $g\left( x \right) = \dfrac{1}{2}x + 1$ .
We know that the graph of 2 functions will intersect only at the points at which they have equal values.
The given graph is intersecting at 2 points. It is also clear from the graph that the function will not intersect at any other points. So, we can say that the functions are equal at only 2 points.
So, option A is correct and B is wrong.
Now we can check whether a function is greater than the other.
We know that if a function is greater than other, then the graph of one function is always above the other. But the graph of the given function intersects and lies both above and below the functions. So, neither function is greater than one another.
So, we can say that options C and D are incorrect.
We know that the graph of a function and its inverse will be symmetric to the line $x = y$ . It is clear from the graph that they are not symmetric. So, one function is not the inverse of the other.
Therefore, we can say that option E is also incorrect.

So, the only relation that is true is option A.

Note:
Alternate solution is given by,
We have the 2 functions as functions $f\left( x \right) = {\left( {x - 3} \right)^2} + 2$ and $g\left( x \right) = \dfrac{1}{2}x + 1$ .
When 2 functions are equal, we can write,
$f\left( x \right) = g\left( x \right)$
$\Rightarrow {\left( {x - 3} \right)^2} + 2 = \dfrac{1}{2}x + 1$
On expanding the square, we get,
$\Rightarrow {x^2} - 6x + 9 + 2 = \dfrac{1}{2}x + 1$
On simplification, we get,
$\Rightarrow {x^2} - 6x - \dfrac{1}{2}x + 10 = 0$
On adding like terms, we get,
$\Rightarrow {x^2} - \dfrac{{13}}{2}x + 10 = 0$
We can multiply throughout with 2.
$\Rightarrow 2{x^2} - 13x + 20 = 0$
Now we can find the discriminant of the quadratic equation to find the number of solutions.
$D = {b^2} - 4ac$
On substituting the values, we get,
$\Rightarrow D = {\left( { - 13} \right)^2} - 4 \times 20 \times 2$
On simplification we get,
$\Rightarrow D = 169 - 160$
So, we have,
$\Rightarrow D = 9$
As the discriminant is positive, there are 2 distinct solutions for x.
Therefore, $f\left( x \right) = g\left( x \right)$ for exactly 2 values of x.
So, option A is correct and b is wrong
As $f\left( x \right) = g\left( x \right)$ for exactly 2 values of x, we can say that none of the given functions is greater than or less than the other function. So, options C and D are also incorrect.
One of the functions is linear. We know that the inverse of the linear function is linear. So $f\left( x \right)$ is not the inverse of $g\left( x \right)$ . So, option E is also incorrect.
So, the only relation that is true is option A.