Question

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>

Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.

Accepted Answer

Step 1: Identify the one-to-one functions from the parabola $$ y = x^2 + 1 . $$Since a parabola is symmetric about its vertex, we can split it into two functions: one for the left side and one for the right side. For the right side, where $$ x \geq 0 $$, the function is $$ g_1(x) = x^2 + 1 . $$Step 2: Determine the inverse of $$ g_1(x) . To $$find $$ g_1^{-1}(x) $$, start by setting $$ y = x^2 + 1 $$ and solve for $$ x . $$Rearrange to get $$ x^2 = y - 1 $$, then take the square root: $$ x = \sqrt{y - 1} . $$Thus, $$ g_1^{-1}(x) = \sqrt{x - 1} . $$Step 3: State the domain and range of $$ g_1(x) . $$Since $$ g_1(x) = x^2 + 1 $$ and we are considering $$ x \geq 0 $$, the domain of $$ g_1(x)  is$$ $$[0, \infty). $$The range is $$[1, \infty)$$ because the smallest value of $$ x^2 + 1  is 1 $$when $$ x = 0 . $$Step 4: State the domain and range of $$ g_1^{-1}(x) . $$The domain of $$ g_1^{-1}(x) = \sqrt{x - 1}  is$$ $$[1, \infty)$$ because the expression under the square root, $$ x - 1 $$, must be non-negative. The range is $$[0, \infty)$$ because the square root function outputs non-negative values. Step 5: Verify consistency with the graph. Check that the domain and range of $$ g_1(x) $$ and $$ g_1^{-1}(x) $$ match the sections of the parabola and its inverse on the graph. Ensure that $$ g_1(x) $$ covers the right side of the parabola and $$ g_1^{-1}(x) $$ reflects this section across the line $$ y = x .$$

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>

Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.

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Key Concepts

One-to-One Functions

Inverse Functions

Domain and Range

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.