Questions: Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. g(x) = x^2 + 1

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. g(x) = x^2 + 1
Transcript text: Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. \[ g(x)=x^{2}+1 \]
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Solution

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Solution Steps

Step 1: Identify the Basic Function

The given function is \( g(x) = x^2 + 1 \). The basic function here is \( f(x) = x^2 \), which is a standard parabola opening upwards with its vertex at the origin (0,0).

Step 2: Apply the Transformation

The function \( g(x) = x^2 + 1 \) is obtained by shifting the basic function \( f(x) = x^2 \) vertically upwards by 1 unit. This means every point on the graph of \( f(x) = x^2 \) is moved 1 unit up.

Step 3: Determine the Domain and Range
  • Domain: The domain of \( g(x) = x^2 + 1 \) is all real numbers, since there are no restrictions on the values that \( x \) can take. Therefore, the domain is \( (-\infty, \infty) \).
  • Range: The range of \( g(x) = x^2 + 1 \) is all values \( y \) such that \( y \geq 1 \). This is because the minimum value of \( x^2 \) is 0, and adding 1 shifts the minimum value to 1.

Final Answer

  • Domain: \( (-\infty, \infty) \)
  • Range: \( [1, \infty) \)
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