/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Let \(f(x)=(x-5)^{2} .\) Find \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) find the vertex and the axis of symmetry and (b) graph the function. $$ f(x)=x^{2}+8 x+20 $$

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

Trajectory of a Flare. The height above the ground of a launched object is a quadratic function of the time that it is in the air. Suppose that a flare is launched from a cliff 64 ft above sea level. If 3 sec after being launched the flare is again level with the cliff, and if 2 sec after that it lands in the sea, what is the maximum height that the flare will reach?

For each of the following, graph the function, label the vertex, and draw the axis of symmetry. $$ f(x)=2(x+7)^{2} $$

Composting. A rectangular compost container is to be formed in a corner of a fenced yard, with 8 ft of chicken wire completing the other two sides of the rectangle. If the chicken wire is \(3 \mathrm{ft}\) high, what dimensions of the base will maximize the container's volume?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.