(a) Given $$ f(x)=2 x^{2} $$. (1) Restrict the domain of $$ f $$ in two different ways so that the inverse is a (2) Hence sketch the graph of $$ f $$ and its inverse for each restriction of the (3) Write the equation of the inverse function in the form $$ f^{-1}(x)=\ldots $$

Question

UpStudy AI · Accepted Answer

We are given the function

f(x) = 2x².

Because f is not one-to-one on all of ℝ (since, for example, f(2) = 8 = f(–2)), we must restrict the domain to obtain an inverse function. One common method is to split the parabola into its two "halves."

────────────────────────────────────────────────────────
Step (1): Choosing Two Different Domain Restrictions

Option 1: Restrict to x ≥ 0.
 • Domain: [0, ∞)
 • On this interval, f is increasing and one-to-one.
 • The range is f([0, ∞)) = [0, ∞).

Option 2: Restrict to x ≤ 0.
 • Domain: (–∞, 0]
 • On this interval, f is decreasing and one-to-one.
 • The range is again f((–∞, 0]) = [0, ∞).

────────────────────────────────────────────────────────
Step (2): Sketching the Graphs

For both options, the graph of y = f(x) = 2x² is a parabola opening upward with vertex (0, 0).

• When the domain is [0, ∞):
 – We only consider the right half of the parabola (x ≥ 0).
 – Its inverse will be a function defined on [0, ∞) that is the reflection of this half about the line y = x.
 – The sketch of f^–1 will be the top-right half of the sideways opening parabola.

• When the domain is (–∞, 0]:
 – We only consider the left half of the parabola (x ≤ 0).
 – Its inverse will similarly be the reflection of this half of the graph about y = x.
 – The inverse graph will lie in the region where the outputs are ≤ 0 (since it “undoes” a negative input).

Recall: The graph of an inverse function is the reflection of the original function’s graph over the line y = x. When graphing, plot y = f(x) and then reflect it about y = x to produce y = f^–1(x).

────────────────────────────────────────────────────────
Step (3): Finding the Inverse Function Equations

To find the inverse, start with

y = 2x².

Solve for x (remembering that the sign of x depends on the selected domain).

1. Solve for x:
  x² = y/2  ⟹  x = ±√(y/2).

Now, choose the correct sign for each restriction.

Option 1: For x ≥ 0,
  x = √(y/2).
Now, swap x and y to write the inverse function:
  f^–1(x) = √(x/2),  with x in [0, ∞).

Option 2: For x ≤ 0,
  x = –√(y/2).
Swap x and y:
  f^–1(x) = –√(x/2),  with x in [0, ∞).

────────────────────────────────────────────────────────
Summary of Answers

(a) (1) Two possible domain restrictions that make f one-to-one are:
  • Domain: [0, ∞)  or  • Domain: (–∞, 0].

(2) For each restriction, the graph of f is just the parabola y = 2x² but with only the right half (x ≥ 0) or left half (x ≤ 0). The graph of the inverse is obtained by reflecting the restricted graph of f over the line y = x.

(3) The equations for the inverse functions are:
  • If f is restricted to [0, ∞):  f^–1(x) = √(x/2)
  • If f is restricted to (–∞, 0]:  f^–1(x) = –√(x/2)

These provide the required answers.
To find the inverse of $$ f(x) = 2x^2 $$, restrict the domain to either $$ x \geq 0 $$ or $$ x \leq 0 $$. For each restriction: 1. **Domain Restriction:** - **Option 1:** $$ x \geq 0 $$ - **Option 2:** $$ x \leq 0 $$ 2. **Inverse Function:** - **For $$ x \geq 0 $$:** $$ f^{-1}(x) = \sqrt{\frac{x}{2}} $$ - **For $$ x \leq 0 $$:** $$ f^{-1}(x) = -\sqrt{\frac{x}{2}} $$ 3. **Graphs:** - The original function $$ f(x) = 2x^2 $$ is a parabola opening upwards with vertex at (0,0). - When restricted to $$ x \geq 0 $$, the inverse is the right half of the parabola reflected over the line $$ y = x $$. - When restricted to $$ x \leq 0 $$, the inverse is the left half of the parabola reflected over the line $$ y = x $$. These restrictions ensure that $$ f $$ is one-to-one, allowing for valid inverse functions.

(a) Given \( f(x)=2 x^{2} \). (1) Restrict the domain of \( f \) in two different ways so that the inverse is a (2) Hence sketch the graph of \( f \) and its inverse for each restriction of the (3) Write the equation of the inverse function in the form \( f^{-1}(x)=\ldots \)

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Calculus Questions