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Chapter 0: Problem 91

Find the domain and range for the function \(f(x)=x^{2}+1\).

Short Answer

Expert verified
Domain: \[ (-\infty, \infty) \]; Range: \[ [1, \infty) \].

Step by step solution

01

Identify the type of function

Recognize that the given function is a quadratic function of the form \[ f(x) = x^2 + 1 \]. Quadratic functions have specific characteristics that will help in determining the domain and range.
02

Determine the domain

The domain of a function is the set of all possible input values (x-values) that can be used. For a quadratic function, any real number can be an input value because there are no restrictions such as division by zero or square roots of negative numbers. Therefore, the domain is all real numbers, which can be written as \[ (-\infty, \infty) \].
03

Determine the range

The range of a function is the set of all possible output values (y-values). For \[ f(x) = x^2 + 1 \], the minimum value of \[ x^2 \] is 0 (when \[ x = 0 \]). Adding 1 to this minimum value gives 1, which is the smallest value that \[ f(x) \] can take. Since \[ x^2 \] can become arbitrarily large for large absolute values of x, the maximum value of \[ f(x) \] is infinity. Therefore, the range is \[ [1, \infty) \].

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial function with the form
where
The expression are linear functions,
and it graphs as a parabola.
In this case, our function is: since it follows the form of , this indicates that it's a quadratic function.
Quadratic functions have a distinct parabolic (U-shaped) graph.
The parabola opens upwards since the coefficient of the term is positive (which is 1 in this function).
domain
The domain of a function refers to all possible input values (x-values) that the function can accept.
For quadratic functions like ,
the domain includes all real numbers because there are no restrictions such as division by zero or the square root of negative numbers.
Another way to think about this is any x-value you plug into the function will produce a valid output, or y-value.
So, the domain for all quadratic functions, including

is
often written in interval notation as
range
The range of a function is the set of all possible output values (y-values) that it can produce.
To determine the range of the quadratic function we examine its graph.
In this quadratic function, the term always produces non-negative results (since squaring any real number yields a positive result or zero).
The smallest value that can be is 0 (when ).
Adding 1 to this minimum value gives us 1,
which is the lowest point on the graph of
The parabola opens upwards,
meaning the values can increase without bound as x moves away from 0 in either direction.
Therefore, the range of is which is often given in interval notation as .

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