Given the function:
Determine for which values of x the following is true: f\left(x\right) > 0
Given the function:
\( y=-x^{2}+49 \)
Determine for which values of x the following is true: \( f\left(x\right) > 0 \)
Given the function:
\( y=x^2+16 \)
Determine for which values of x \( f(x) < 0 \) holds
Given the function:
\( y=x^2+16 \)
Determine for which values of x is \( f(x) > 0 \) true
Given the function:
\( y=2x^2+16 \)
Determine for which values of x the following holds: \( f(x) < 0 \)
Given the function:
\( y=2x^2+6 \)
Determine for which values of x is \( f\left(x\right) > 0 \) true
Given the function:
Determine for which values of x the following is true: f\left(x\right) > 0
To determine for which values of the function is positive, we solve the inequality:
We start by finding the roots of the associated quadratic equation:
This can be rewritten as:
Solving for , we have:
The roots are and . These points are where the parabola intersects the x-axis. Since the parabola opens downwards (as the coefficient of is negative), the function is positive between the roots.
Therefore, the function over the interval:
Thus, the correct choice is:
-7 < x < 7
Given the function:
Determine for which values of x f(x) < 0 holds
We start by considering the function given: . Our task is to find the values of making , i.e., .
Let's analyze the expression:
Thus, the expression is always at least 16, and there are no values for which .
The solution is that there are No x values where .
Hence, option 4 is correct: No x.
No x
Given the function:
Determine for which values of x is f(x) > 0 true
To solve the problem of determining for which values of the function is positive, we proceed as follows:
Step 1: Analyze the function .
The expression is non-negative (i.e., ) for all real numbers . Therefore, the smallest value can take is 0.
Step 2: Evaluate the function at its minimum value.
Substituting the minimum value of into the function gives us:
.
Step 3: Determine for which the function is positive.
Since the minimum value of the function is 16, which is greater than 0, the function is greater than 0 for all real numbers .
Thus, the solution to the problem is that the function is positive for all .
All x
Given the function:
Determine for which values of x the following holds: f(x) < 0
To solve this problem, consider the function . Our objective is to find values of for which .
The function, , is a quadratic function in standard form. Here, , , and .
Since the discriminant is negative, the function has no real roots, confirming that the quadratic function never takes a value below zero. The vertex is at , which is above zero, indicating all function values are positive.
Therefore, the function is never less than zero, implying that there are no values of for which .
Hence, the solution is No x.
No x
Given the function:
Determine for which values of x is f\left(x\right) > 0 true
To solve this problem, let's analyze the quadratic function :
Therefore, the solution to the problem is all x.
All x
Given the function:
\( y=3x^2+21 \)
Determine for which values of x is \( f\left(x\right) < 0 \) true
Given the function:
\( y=3x^2+21 \)
Determine for which values of x the following holds:
\( f\left(x\right) > 0 \)
Given the function:
\( y=4x^2+100 \)
Determine for which values of x the following holds:
\( f\left(x\right) < 0 \)
Given the function:
\( y=4x^2+100 \)
Determine for which values of x the following holds:
\( f\left(x\right) > 0 \)
Given the function:
\( y=-x^2-4 \)
Determine for which values of x the following is true:
\( f\left(x\right) < 0 \)
Given the function:
Determine for which values of x is f\left(x\right) < 0 true
To solve the given problem, follow these steps:
Given the function is always positive and never reaches zero or becomes negative, there are no values of for which .
Thus, the solution to this problem is No x.
No x
Given the function:
Determine for which values of x the following holds:
f\left(x\right) > 0
To solve the problem, we must determine when the function is positive:
Step 1: Analyze the quadratic function . Since and , the parabola opens upwards with vertex at .
Step 2: Compute the function's value at the vertex. For , , which is positive.
Step 3: Understand the behavior for any . Since is non-negative and added ensures , the entire graph of lies above the x-axis.
Therefore, the solution is that for all values of , the function is always greater than 0.
The correct answer is All x.
All x
Given the function:
Determine for which values of x the following holds:
f\left(x\right) < 0
To solve for the values of where , follow these steps:
Since the minimum value of the function is , which is greater than zero, the function never reaches below zero. Hence, there are no values for which the function .
Therefore, the solution is No x.
No x
Given the function:
Determine for which values of x the following holds:
f\left(x\right) > 0
To determine for which values of the function is greater than 0, we analyze the structure of the quadratic equation:
The function is , where is always non-negative for any real number because squaring any real number gives a non-negative result, and multiplying by a positive constant (4) remains non-negative.
The constant term in the function is , which is positive. Therefore, the smallest value can take is 0 (when ), making the minimum value of the function .
Since is always greater than , for all values of , this quadratic function never reaches or becomes negative.
In conclusion, the function is positive for all values of .
Therefore, the solution to the problem is All .
All x
Given the function:
Determine for which values of x the following is true:
f\left(x\right) < 0
To solve this problem, we'll follow these steps:
Now, let's solve it:
Step 1: Consider the given quadratic function . The function represents a downward-opening parabola due to the negative sign before . The entire graph of the parabola, being shifted downward by 4 units with the term , will be wholly beneath the x-axis since there's no positive vertex or value. The vertex of the parabola is at , which is already below zero.
Step 2: Because the quadratic term causes the graph to be a parabola opening downwards, it means that for any , , which is always less than zero. Thus, the inequality is satisfied for all real numbers .
Therefore, the solution is that the inequality is true for all .
All x
Given the function:
\( y=-x^2-4 \)
Determine for which values of x the following holds:
\( f\left(x\right) > 0 \)
Given the function:
\( y=-3x^2-9 \)
Determine for which values of x the following holds:
\( f\left(x\right) < 0 \)
Look at the function below:
\( y=-3x^2-9 \)
Determine for which values of \( x \) the following is true:
\( f(x) > 0 \)
Look at the function below:
\( y=-4x^2-12 \)
Then determine for which values of \( x \) the following is true:
\( f\left(x\right) < 0 \)
Look at the function below:
\( y=-4x^2-12 \)
Determine for which values of \( x \) the following is true:
\( f\left(x\right) > 0 \)
Given the function:
Determine for which values of x the following holds:
f\left(x\right) > 0
To solve this problem, we'll follow these steps:
Step 1: The function given is , which is a parabola opening downwards because of the negative coefficient of .
Step 2: The vertex of this parabola occurs at , considering no term present (i.e., ). The vertex value of is .
Step 3: The vertex, being the highest point at , means the entire parabola remains below the x-axis.
Since the maximum point is negative and the parabola only descends from this vertex, the function remains less than zero for every . There are no values making positive.
Therefore, the solution is No x.
No x
Given the function:
Determine for which values of x the following holds:
f\left(x\right) < 0
Given the quadratic function , we want to determine when .
First, observe that the function is a downward-opening parabola because the coefficient of is negative (). This means the parabola opens downwards.
To find when the parabola is below the x-axis (), we should first check whether there are any real roots, since this implies crossing the x-axis.
The function is reformulated as:
.
To find the roots, rearrange and solve:
or .
Since yields no real solutions (as no real number squared equals a negative), there are no x-intercepts.
This indicates the parabola does not cross the x-axis and is entirely below it (since it opens downward and has no real roots).
Thus, the function for all real values of .
Therefore, the condition is satisfied for all x, meaning the correct choice is:
All x
.All x
Look at the function below:
Determine for which values of the following is true:
f(x) > 0
To solve this problem, let's analyze the function and determine when it is greater than zero.
The function is a quadratic equation of the form where , , and .
Our task is to find when .
Step 1: Rewrite the inequality:
Step 2: Add 9 to both sides to isolate the quadratic term:
Step 3: Divide through by -3 (note that dividing by a negative flips the inequality sign):
Step 4: Analyze
A square of a real number is always non-negative, meaning . Therefore, is impossible since there are no real values of such that the square of is a negative number.
Conclusion: The inequality has no real solutions. Therefore, no values of satisfy the inequality.
The correct answer is that no values of will make .
No values of
Look at the function below:
Then determine for which values of the following is true:
f\left(x\right) < 0
Let's solve this step-by-step:
Given that the parabola opens downwards and never meets the x-axis, the function is always less than zero for all real .
Therefore, the solution to the problem is that the function is negative for all values of .
All values of
Look at the function below:
Determine for which values of the following is true:
f\left(x\right) > 0
The goal is to find the values of for which given . Start by analyzing the equation .
Since the quadratic term is negative (), the parabola opens downwards. This means the maximum point of the parabola (its vertex) is at the top.
Find the vertex using the formula for the -coordinate of the vertex, . Here, , so the vertex is at .
Calculate -value at the vertex:
.
This evaluation confirms , which is less than 0 at the vertex.
Since the entire parabola opens downward and the highest point achieves is still negative (), the function is never greater than 0 at any point.
No values satisfy . Therefore, no values of make the quadratic positive.
Thus, the answer is: No values of .
No values of
Look at the function below:
\( y=-5x^2-10 \)
Determine for which values of \( x \) the following is true:
\( f\left(x\right) < 0 \)
Look at the function below:
\( y=-5x^2-10 \)
Then determine for which values of \( x \) the following is true:
\( f\left(x\right) > 0 \)
Look at the function below:
\( y=x^2-4 \)
Then determine for which values of \( x \) the following is true:
\( f\left(x\right) < 0 \)
Look at the function below:
\( y=x^2-4 \)
Then determine for which values of \( x \) the following is true:
\( f(x) > 0 \)
Look at the following function:
\( y=3x^2-27 \)
Determine for which values of \( x \) the following is true:
\( f\left(x\right) < 0 \)
Look at the function below:
Determine for which values of the following is true:
f\left(x\right) < 0
Let's solve this step-by-step.
Therefore, the solution to the problem is that for all values of .
All values of
Look at the function below:
Then determine for which values of the following is true:
f\left(x\right) > 0
To solve this problem, we need to analyze the quadratic function given by:
1. Step 1: Find the vertex of the parabola.
The formula for a quadratic function is . For our function, , , .
Since , the x-coordinate of the vertex is .
2. Step 2: Calculate the y-coordinate of the vertex.
Substitute into the function:
.
3. Step 3: Analyze the parabola.
The vertex is at (0, -10). Since the vertex itself is below the x-axis and the parabola opens downwards (given by a < 0 ), the entire parabola is below the x-axis.
As a result, there are no values of for which the function is greater than 0.
Therefore, the correct answer to the problem is that there are no values of .
No values of
Look at the function below:
Then determine for which values of the following is true:
f\left(x\right) < 0
To determine for which values of the function is less than 0, we need to solve the inequality .
First, find when the function equals zero by solving . This gives the roots as , or , specifically and .
Next, perform a sign analysis of in the intervals defined by these roots: , , and .
Thus, the function is less than 0 for .
The correct interval representing the solution is .
-2 < x < 2
Look at the function below:
Then determine for which values of the following is true:
f(x) > 0
To find the solution to when is greater than zero, we start by analyzing the inequality:
This expression can be factored as:
The roots of the equation are and . These points divide the real number line into intervals. We need to determine on which intervals the expression is positive. Let's analyze the intervals:
Therefore, the function is positive for or .
Thus, the solution to the inequality is:
or .
x > 2 or x < -2
Look at the following function:
Determine for which values of the following is true:
f\left(x\right) < 0
To solve the problem of finding the values of for which , we start by solving the equation :
Step 1: Solve the equation .
The solutions and are the roots of the quadratic function. This means the function transitions from negative to non-negative (and vice versa) at these points.
Step 2: Analyze the intervals defined by the roots.
Since the quadratic is a parabola opening upwards (coefficient of is positive), the function will be negative between the roots.
Therefore, check the interval :
The function is negative in the interval .
Thus, the values of for which are .
-3 < x < 3