Problem 46

Question

Find the domain and range of the function. $$f(x)=x^{2}-2 x-3$$

Step-by-Step Solution

Verified

Answer

Domain: all real numbers; Range: $y \geq -4$.

1Step 1: Identify the Type of Function

The function given is a quadratic function, which has the general form $f(x) = ax^2 + bx + c$. In our case, it is given by $f(x) = x^2 - 2x - 3$. Quadratic functions represent parabolas when graphed.

2Step 2: Determine the Domain

For any quadratic function, the domain includes all real numbers because the function is defined for every real number. This means that you can plug any real number into the function and receive a valid output.

3Step 3: Analyze the Range

A quadratic function can open upwards or downwards depending on the sign of the coefficient of the $x^2$ term. Here, the coefficient is positive ($+1$), so the parabola opens upwards. The vertex will be the minimum point of the function.

4Step 4: Find the Vertex

To find the vertex of the parabola, we use the formula for the x-coordinate: $-\frac{b}{2a}$. With $a = 1$ and $b = -2$, the x-coordinate is $x = 1$. Substitute this back into the function to find the y-coordinate: $f(1) = 1^2 - 2(1) - 3 = -4$. Thus, the vertex is $(1, -4)$.

5Step 5: Determine the Range

Since the parabola opens upwards and the vertex is at its lowest point, the range consists of all real numbers greater than or equal to the y-coordinate of the vertex. Thus, the range of the function is $y \geq -4$.

Key Concepts

Domain of a FunctionRange of a FunctionParabolasVertex of a Parabola

Domain of a Function

The domain of a quadratic function refers to all the possible x-values that you can input into the function. For the function $f(x) = x^2-2x-3$, which is a quadratic function, the domain is quite simple to determine. Because quadratic functions can accept any real number as input, their domain includes all real numbers from negative infinity to positive infinity.

Quadratic functions are defined for every real number without restrictions.
This means that whatever real number is plugged into the function, it will produce a valid output.

So, for our function $f(x) = x^2-2x-3$, the domain is all real numbers, often written as $x \in (-\infty, \infty)$.

Range of a Function

The range of a function is all the possible y-values, or outputs, that the function can produce. For a quadratic function like $f(x) = x^2-2x-3$, the range depends on whether the parabola opens upwards or downwards. Since the coefficient of the $x^2$ term is positive (+1), it tells us the parabola opens upwards.

For upward-opening parabolas, the range is all the y-values greater than or equal to the vertex's y-coordinate.
The lowest point on the graph of this function is the vertex.

Hence, for $f(x) = x^2-2x-3$, the range includes all real numbers $y$ such that $y \geq -4$, since -4 is the y-coordinate of the vertex. This can be expressed as $y \in [-4, \infty)$.

Parabolas

A parabola is the graph of a quadratic function, which has the form $f(x) = ax^2 + bx + c$. Parabolas are curved and symmetrical shapes that open either upwards or downwards based on the sign of $a$.

If $a > 0$, the parabola opens upwards.
If $a < 0$, it opens downwards.

This symmetry means parabolas have a line of symmetry, which passes through their vertex. For our function $f(x) = x^2-2x-3$, we have an upwards-opening parabola because $a = 1$, which is positive. Understanding the shape and direction of parabolas is crucial because it influences the function's range and helps in identifying key points like the vertex.

Vertex of a Parabola

The vertex of a parabola is a significant point because it represents either the maximum or minimum point based on whether the parabola opens downwards or upwards, respectively. To find the vertex of a parabola represented by $f(x) = ax^2 + bx + c$, use the formula for the x-coordinate of the vertex: $-\frac{b}{2a}$.

For $f(x) = x^2-2x-3$, plug $a = 1$ and $b = -2$ into the formula, giving an x-coordinate of $x = 1$.
To find the y-coordinate, substitute $x = 1$ back into the function: $f(1) = 1^2 - 2 \times 1 - 3 = -4$.

Thus, the vertex of the parabola is $(1, -4)$, indicating the lowest point on the graph because the parabola opens upwards. The position of the vertex helps determine the range and provides a clear depiction of the function's graph.

Problem 46

Other exercises in this chapter

Problem 46

Find all zeros of the polynomial. $$P(x)=x^{3}-7 x^{2}+17 x-15$$

View solution

Problem 46

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{6}-3 x^{5}-13 x^{4}+29 x^{3}-27 x^{2}+32 x-12$$

View solution

Problem 46

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ in large and small viewing rectangles, as in Example $3(b)$. $$P(x)=2 x^{2}-x^{12}

View solution

Problem 46

Use synthetic division and the Remainder Theorem to evaluate $P(c)$. $$P(x)=6 x^{5}+10 x^{3}+x+1, \quad c=-2$$

View solution