Solve f(x)=x² When f(6)=6+a: Finding the Missing Value

Function Evaluation with Algebraic Equations

Complete:

the value a a The subtraction of the function point:

f(x)=x2 f(x)=x^2

f(6)=6+a f(6)=6+a

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find A
00:04 Substitute appropriate values according to the given data, and solve for A
00:16 Calculate the exponent
00:26 Isolate the unknown A
00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Complete:

the value a a The subtraction of the function point:

f(x)=x2 f(x)=x^2

f(6)=6+a f(6)=6+a

2

Step-by-step solution

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

  • Step 1: Evaluate f(6) f(6) using the function f(x)=x2 f(x) = x^2 .
  • Step 2: Set f(6)=6+a f(6) = 6 + a and solve for a a .

Now, let's apply these steps:
Step 1: Compute f(6) f(6) . Since f(x)=x2 f(x) = x^2 , we have f(6)=62=36 f(6) = 6^2 = 36 .
Step 2: We are given that f(6)=6+a f(6) = 6 + a . Therefore, substitute the evaluated value:
36=6+a 36 = 6 + a .

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

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

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

3

Final Answer

a=30 a=30

Key Points to Remember

Essential concepts to master this topic
  • Function Rule: Substitute input value into function formula first
  • Technique: Calculate f(6) = 6² = 36, then solve 36 = 6 + a
  • Check: Verify f(6) = 36 equals 6 + 30 = 36 ✓

Common Mistakes

Avoid these frequent errors
  • Setting up equation incorrectly
    Don't write a = 6² = 36 directly! This skips the crucial step of using the given condition f(6) = 6 + a. Always evaluate the function first, then set it equal to the given expression to solve for the unknown.

Practice Quiz

Test your knowledge with interactive questions

Complete:

The missing value of the function point:

\( f(x)=x^2 \)

\( f(?)=16 \)

FAQ

Everything you need to know about this question

Why can't I just say a = 36 since f(6) = 36?

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Because the problem states that f(6) = 6 + a, not f(6) = a. You need to use this condition: since f(6) = 36 and f(6) = 6 + a, then 36 = 6 + a, so a = 30.

What does f(6) actually mean?

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f(6) means the output when x = 6. For f(x) = x², substitute 6 for x: f(6) = 6² = 36. It's the value of the function at that input.

How do I solve 36 = 6 + a for a?

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Use inverse operations! Since 6 is added to a, subtract 6 from both sides: 36 - 6 = a, so a = 30.

Can I check my answer somehow?

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Yes! Substitute back: if a = 30, then 6 + a = 6 + 30 = 36. Since f(6) = 36 too, both sides match. Your answer is correct!

What if the function was different, like f(x) = x³?

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Same method! Calculate f(6) = 6³ = 216, then solve 216 = 6 + a to get a = 210. The process stays the same regardless of the function.

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Parabola of the form y=x²