Problem 31 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=4-x^{2} $$

Step-by-Step Solution

Answer

The range of the function is \((-\infty, 4]\); the domain is \((-\infty, \infty)\).

Step 1: Understand the Function

The function given is \( f(x) = 4 - x^2 \). This represents a downward-facing parabola with its vertex at the point \((0, 4)\) because \(-x^2\) indicates a parabola that opens downwards, and the constant \(4\) shifts the vertex to that y-coordinate.

Step 2: Graph the Function

Use a graphing calculator or software to draw the graph of \( f(x) = 4 - x^2 \). The plot will show the vertex at the point (0,4) and the parabola opening downwards, crossing the y-axis at 4 and the x-axis at \(-2\) and \(2\).

Step 3: Determine the Domain

The parabola \( f(x) = 4 - x^2 \) is defined for all real numbers. Therefore, the domain of \( f(x) \) is \( (-\infty, \infty) \). It means you can plug any real number into \( x \) and still get a valid output for \( f(x) \).

Step 4: Determine the Range

To find the range, observe the highest and lowest points on the graph of the function. The highest point (the vertex of the parabola) is at \( y = 4 \), and the parabola extends downwards to \(-\infty\). Thus, the range of \( f \) is \(( -\infty, 4 ]\), including 4 but no values above it.

Key Concepts

Domain and Range

When we talk about the "domain" of a function, we are referring to all the possible input values (x-values) that can be used in the function. In the case of the quadratic function \( f(x) = 4 - x^2 \), there are no restrictions on the x-values; thus, the domain is all real numbers, or \( (-\infty, \infty) \). You can use any real number for \( x \) and the function will still compute.

The "range" of a function, on the other hand, refers to all possible output values (y-values). For our function \( f(x) = 4 - x^2 \), the maximum value occurs at the vertex of the parabola, \( y = 4 \). Since the parabola opens downward, the values of \( y \) decrease towards negative infinity. Therefore, the range of this function is \( ( -\infty, 4 ] \), which includes all y-values below and including 4.

Remember:

• Domain relates to x-values.
• Range relates to y-values.

Vertex of a Parabola

In a quadratic function of the form \( f(x) = ax^2 + bx + c \), the graph results in a parabola. The "vertex" of this parabola is a key point where the curve changes direction, known as the maximum or minimum point. For the function \( f(x) = 4 - x^2 \), the vertex happens to be a maximum point because the parabola opens downwards.

You can determine the vertex of the parabola from the function's equation. Here, since we have \( f(x) = 4 - x^2 \), it can be rewritten as \( f(x) = -(x^2) + 4 \). This indicates a vertex at \( (0, 4) \), with \( 0 \) being the x-coordinate and \( 4 \) the y-coordinate. This poiont represents the peak of the parabola and is important in determining the range of the function.

Some key points about vertices:

• For \( ax^2 \) with \( a > 0 \), the vertex is the minimum point.
• For \( ax^2 \) with \( a < 0 \), the vertex is the maximum point.
• The vertex tells us the extremal value of the function.

Quadratic Equations

Quadratic equations form the basis of quadratic functions like \( f(x) = 4 - x^2 \). These equations generally take the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic equation is always a parabola.

Key characteristics include:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.
• The x-intercepts (roots) can be found by setting \( f(x) = 0 \) and solving for \( x \).

In have a quadratic function like \( f(x) = 4 - x^2 \), the coefficient of \( x^2 \) is \( -1 \) (hence \( a < 0 \)), making it a downward-opening parabola. This makes it easy to find the vertex and assess the function's behavior by examining its maximum and minima.