Parabola of the form y=x²: Complete the missing numbers

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation from the function definition.
• Step 2: Solve the equation by taking the square root of both sides.
• Step 3: Identify all possible values for $x$.
• Step 4: Compare with the given answer choices.

Now, let's work through each step:

Step 1: We start with the equation given by the function $f(x) = x^2$. We know $f(?) = 16$, so we can write:

$x^2 = 16$

Step 2: To solve for $x$, we take the square root of both sides of the equation:

$x = \pm \sqrt{16}$

Step 3: Solve for $\sqrt{16}$:

The square root of 16 is 4, so:

$x = 4$ or $x = -4$

This gives us the two solutions: $x = 4$ and $x = -4$.

Step 4: Compare these solutions to the answer choices. The correct choice is:

$f(4)$ and $f(-4)$

Therefore, the solution to the problem is $f(4)$ and $f(-4)$.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation $x^2 = 9$.
• Step 2: Solve for $x$ by taking the square root of both sides.
• Step 3: Choose the correct answer from the given options.

Now, let's work through each step:
Step 1: We set $x^2 = 9$.
Step 2: Solving for $x$, we take the square root of both sides: $x = \pm \sqrt{9}$.
Step 3: This yields two solutions: $x = 3$ and $x = -3$.

Comparing these values with the given choices:

• Choice 1: $f(3)$ corresponds to $x = 3$.
• Choice 3: $f(-3)$ corresponds to $x = -3$.

Both choices $f(3)$ and $f(-3)$ are correct, leading us to select the combined choice: Answer A+C.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation based on the function $f(x)=x^2$ for $f(?)=25$.
• Step 2: Solve for $x$ by applying the square root operation.

Now, let's work through each step:
Step 1: We start with the equation $x^2 = 25$ derived from $f(x) = 25$.
Step 2: To solve for $x$, we take the square root of both sides:

$x = \pm \sqrt{25}$

Calculating the square root gives us $x = \pm 5$. However, we are looking for a specific point that fits one of the answer choices:
Therefore, the solution based on the choices provided is $x = 5$.

Concluding, the missing value of the function point is $f(5)$, which coincides with choice 1.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation $x^2 = 81$.
• Step 2: Solve for $x$ by taking the square root of both sides.
• Step 3: Confirm that the solutions are integers provided in the choices.

Now, let's work through each step:
Step 1: The problem gives us the function $f(x) = x^2$ and asks us to find values of $x$ such that $f(x) = 81$
Step 2: Solving $x^2 = 81$, we take the square root of both sides to get $x = \pm \sqrt{81}$.
Step 3: Compute $\sqrt{81} = 9$, which gives us $x = 9$ or $x = -9$. Therefore, the solutions are $x = 9$ and $x = -9$.

Considering the given choices, we can identify that $f(-9) = 81$, which corresponds to choice 1 in the problem.

Therefore, the solution to the problem is $\boldsymbol{f(-9)}$.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation based on the function and given value.
• Step 2: Solve the quadratic equation considering both positive and negative roots.

Now, let's work through each step:

Step 1: Set up the equation based on the given condition. We know that
$f(x) = x^2$ and we need $f(?) = 64$. So we equate:
$x^2 = 64$.

Step 2: Solve for $x$ using the square root rule, which tells us that if $x^2 = a$, then $x = \pm\sqrt{a}$.

Applying this to our equation:
$x = \pm\sqrt{64}$.
Calculate the square root: $\sqrt{64} = 8$.
Therefore, the solutions are $x = 8$ and $x = -8$.

Thus, we have $f(8) = 64$ and $f(-8) = 64$.

Among the given choices, $f(8)$ and $f(-8)$ is the correct choice.

Therefore, the missing value is $f(8)$ and $f(-8)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given equation from the function.
• Step 2: Use the square root method to find possible values of $x$.
• Step 3: Compare solutions with the provided answer choices.

Now, let's work through each step:
Step 1: Start with the equation $f(x) = 49$ which gives us $x^2 = 49$.
Step 2: Solve for $x$ by taking the square root of both sides, which leads to two possible solutions: $x = \sqrt{49}$ and $x = -\sqrt{49}$. Thus, $x = 7$ or $x = -7$.
Step 3: Compare these solutions with the answer choices:

• $f(7)$
• $f(-7)$
• $f(-3)$ (this is incorrect as $f(-3) = 9$)
• Answers a + b

The correct answers based on the solutions are $f(7)$ and $f(-7)$, making choice 4, "Answers a + b," the correct option.

Therefore, the correct answer is: Answers a + b.

To solve this problem, we'll follow these steps:

• Step 1: Calculate $f(4)$ using the given function.
• Step 2: Set $f(4)$ equal to $6 + a$ and solve for $a$.

Now, let's work through each step:

Step 1: The function given is $f(x) = x^2$. We want to find $f(4)$, which is:

$f(4) = 4^2 = 16$

Step 2: According to the problem, $f(4) = 6 + a$. Therefore, we set:

$16 = 6 + a$

Subtract 6 from both sides to solve for $a$:

$16 - 6 = a$

$a = 10$

Therefore, the value of $a$ is $a = 10$.

To solve this problem, we will follow a methodical approach:

• Step 1: Evaluate $f(6)$ using the function $f(x) = x^2$.
• Step 2: Set $f(6) = 6 + a$ and solve for $a$.

Now, let's apply these steps:
Step 1: Compute $f(6)$. Since $f(x) = x^2$, we have $f(6) = 6^2 = 36$.
Step 2: We are given that $f(6) = 6 + a$. Therefore, substitute the evaluated value:
$36 = 6 + a$.

To find $a$, solve the equation $36 = 6 + a$. Subtract $6$ from both sides:

$36 - 6 = a$ which simplifies to $a = 30$.

Therefore, the solution to the problem is $a = 30$.