Find when and . (1 point) Responses Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x squared plus 6 x plus 9 over 2 x squared End Fraction Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction Start Root x plus 3 End Root over 2 x squared End Fraction Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x plus 3 over 2 x squared End Fraction
The expression given is (f∘g)(x) = (x^2 + 6x + 9) / (2x^2).
To find when (f∘g)(x) = 0, we need to set the expression equal to zero and solve for x:
(x^2 + 6x + 9) / (2x^2) = 0
To solve this equation, we can first multiply both sides by 2x^2 to eliminate the denominator:
x^2 + 6x + 9 = 0
Next, we can factor the quadratic equation:
(x + 3)(x + 3) = 0
(x + 3)^2 = 0
To find the value of x that makes the expression equal to zero, we can take the square root of both sides:
√((x + 3)^2) = √0
x + 3 = 0
x = -3
So, the expression (f∘g)(x) = 0 when x = -3.
To find the points of intersection between the functions and expressions provided, we need to find the values of x for which the expressions are equal.
1. First expression: f.g(x) = (x^2 + 6x + 9) / (2x^2)
2. Second expression: f.g(x) = (√(x+3)) / (2x^2)
To find the points of intersection, we need to set these two expressions equal to each other and solve for x:
(x^2 + 6x + 9) / (2x^2) = (√(x+3)) / (2x^2)
To simplify the equation, let's multiply both sides by 2x^2 to eliminate the denominators:
2(x^2 + 6x + 9) = √(x+3)
Now, we need to square both sides of the equation to eliminate the square root:
4(x^2 + 6x + 9)^2 = x + 3
Expanding the squared term:
4(x^4 + 12x^3 + 54x^2 + 108x + 81) = x + 3
Simplifying:
4x^4 + 48x^3 + 216x^2 + 432x + 324 = x + 3
Rearranging the terms to get a quadratic equation:
4x^4 + 48x^3 + 216x^2 + 431x + 321 = 0
Now, we need to solve this equation for x. However, finding the exact solutions for a quartic equation can be complex and may require numerical methods. So, we can utilize software or calculators to approximate the solutions.
To find the points where the two functions (denoted as f.g(x)) intersect, we need to set them equal to each other and solve for x.
1. First, let's set up the equation:
f.g(x) = (x^2 + 6x + 9) / (2x^2)
f.g(x) = √(x + 3) / (2x^2)
2. Now, we can equate these two expressions and solve for x:
(x^2 + 6x + 9) / (2x^2) = √(x + 3) / (2x^2)
3. To solve this equation, let's first eliminate the denominators by multiplying both sides by (2x^2):
(x^2 + 6x + 9) = √(x + 3)
4. Square both sides of the equation to eliminate the square root:
(x^2 + 6x + 9)^2 = (x + 3)
5. Expand the left side of the equation:
x^4 + 12x^3 + 45x^2 + 54x + 81 = x + 3
6. Rearrange the equation to form a quadratic equation:
x^4 + 12x^3 + 45x^2 + 54x + 81 - x - 3 = 0
x^4 + 12x^3 + 45x^2 + 53x + 78 = 0
7. The equation is now in the form ax^4 + bx^3 + cx^2 + dx + e = 0. To find the values of x, you can either solve it algebraically or use numerical methods such as factoring, synthetic division, or graphing calculators.
Note: The steps provided above outline the general process of solving the equation for x symbolically. However, the equation might not have an algebraic solution, in which case numerical methods would be required.