Question 3

Function $f$ is defined as $f(x) = 2x + 5$ and $(f \circ g)(x) = 2x^2 - 6x + 9$ .

How many of the following four statements are true based on the information above?

$g(1) = 0$ .
Function $g$ intersects the x-axis at exactly one point.
$f^{-1}(x) = \frac{1}{2}x + \frac{5}{2}$ .
$f(x) + g(x) = x^2 - x + 7$ .

Explanation

First, we need to determine the function $g(x)$ .

Given $f(x) = 2x + 5$ and $(f \circ g)(x) = 2x^2 - 6x + 9$ .

Based on the definition of function composition $(f \circ g)(x) = f(g(x))$ , we have:

f(g(x)) = 2(g(x)) + 5

2(g(x)) + 5 = 2x^2 - 6x + 9

2(g(x)) = 2x^2 - 6x + 4

g(x) = x^2 - 3x + 2

Now, let's check the truth of each statement.

Statement 1

The value of $g(1)$ :

g(1) = 1^2 - 3(1) + 2 = 1 - 3 + 2 = 0

Statement $(1)$ is True.

Statement 2

To determine the number of intersection points of the quadratic function $g(x) = x^2 - 3x + 2$ with the x-axis, we check the discriminant ( $D = b^2 - 4ac$ ).

a = 1, b = -3, c = 2

D = (-3)^2 - 4(1)(2) = 9 - 8 = 1

Since $D > 0$ , the function $g$ intersects the x-axis at two distinct points, not one.

Statement $(2)$ is False.

Statement 3

Finding the inverse of $f(x) = 2x + 5$ :

Let $f(x) = y$ , then:

y = 2x + 5

2x = y - 5

x = \frac{y - 5}{2}

x = \frac{1}{2}y - \frac{5}{2}

So the inverse is:

f^{-1}(x) = \frac{1}{2}x - \frac{5}{2}

Statement $(3)$ is False (it should be minus $\frac{5}{2}$ , not plus).

Statement 4

Summing $f(x)$ and $g(x)$ :

f(x) + g(x) = (2x + 5) + (x^2 - 3x + 2)

= x^2 + (2x - 3x) + (5 + 2)

= x^2 - x + 7

Statement $(4)$ is True.

Conclusion

The correct statements are statements $(1)$ and $(4)$ . Thus, there are 2 true statements.

Command Palette

Number 3