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Chapter 11: Problem 39

Let \(f(x)=(x-5)^{2} .\) Find \(x\) such that \(f(x)=16\).

Short Answer

Expert verified
x=9 and x=1 .

Step by step solution

01

Write Down the Given Function

The given function is
02

Set the Function Equal to 16

Set the function equal to 16 and write the equation:
03

Take Square Root of Both Sides

To isolate the term containing x, take the square root of both sides:
04

Solve for x

To fully solve for x, add 5 to both potential solutions: { x

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic functions
Quadratic functions are a type of polynomial that follow the form \(f(x) = ax^2 + bx + c\). These functions create a parabola when graphed, which can open either upwards or downwards.
In your original exercise, the function \(f(x) = (x-5)^2\) is already in a factored form, making it easier to visualize. The expression \((x-5)^2\) tells us that the parabola opens upwards and has its vertex at the point \((5,0)\). Quadratic functions are very symmetrical around their vertex.
Understanding how to handle these functions is crucial, as they form the basis for solving many real-world problems.
square root method
The square root method is a common technique used to solve quadratic equations, especially when they are in the form \(ax^2 = c\). This method involves isolating the squared term and then taking the square root of both sides to solve for \(x\).
In the given exercise, the function \(f(x)\) was equated to \(16\): \((x-5)^2 = 16\).
Taking the square root of both sides, we get two potential solutions:
  • \(x-5 = 4\) and
  • \(x-5 = -4\)
.
This yields our next step in solving for \(x\). Remember, when you square root both sides of an equation, you must consider both the positive and negative roots.
isolating variables
Isolating variables is a key strategy in solving equations. The idea is to get the variable you're solving for by itself on one side of the equation.
It's like peeling away layers until you find what you're looking for.
In the exercise, after taking the square root, we isolated \(x\) by solving:
  • \(x-5=4\)
  • \(x-5=-4\)
.
By adding \(5\) to both sides in each equation, we finally find our solutions:
  • \(x=9\)
  • \(x=1\)
.
function notation
Function notation is a concise way to represent functions. Typically, it's written as \(f(x)\). This signifies that \(f\) is a function of \(x\).
In the given exercise, \(f(x) = (x-5)^2\), means whenever you input \(x\) into the function, you subtract \(5\) from \(x\), then square the result.
The exercise asks you to find \(x\) such that \(f(x)=16\), making it crucial to understand that you're seeking \(x\)-values for which this condition holds true.
Using function notation helps us clearly communicate and understand mathematical ideas without ambiguity.

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Most popular questions from this chapter

Solve by completing the square. Show your work. $$ x^{2}+8 x=9 $$

Explain how the leading coefficient of a quadratic function can be used to determine if a maximum or a minimum function value exists.

For each of the following, graph the function and find the vertex, the axis of symmetry, the maximum value or the minimum value, and the range of the function. $$ h(x)=-2(x-1)^{2}-3 $$

Solve. $$ 3 t^{2}-1=6 $$

For each of the following, graph the function and find the maximum value or the minimum value and the range of the function. $$ f(x)=-(x-1)^{2}+3 $$
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