Math / Algebra

QuestionA. -2 B. 2 C. 0 D. $\infty$ E.None Q.33) Let $f(x)=2 x^{2}+3$ find the value of $x$ such that $f^{-1}(x)=2$ A. 19 B. 8 C. -19 D. 2 E.None

Studdy Solution

STEP 1

1. We are given the function $f(x) = 2x^2 + 3$ .
2. We need to find the value of $x$ such that $f^{-1}(x) = 2$ .

STEP 2

1. Understand the meaning of $f^{-1}(x) = 2$ .
2. Set up the equation based on the inverse function.
3. Solve the equation for $x$ .
4. Verify the solution.

STEP 3

Understand that $f^{-1}(x) = 2$ means that when $f(x) = y$ , then $f^{-1}(y) = x$ . Therefore, we need to find $y$ such that $f(y) = 2$ .

STEP 4

Set up the equation $f(y) = 2$ :
$2y^2 + 3 = 2$

STEP 5

Solve the equation for $y$ :
$2y^2 + 3 = 2$ $2y^2 = 2 - 3$ $2y^2 = -1$

STEP 6

Since $2y^2 = -1$ results in a negative number on the right side, there is no real number solution for $y$ .

STEP 7

Verify the solution. Since there is no real solution for $y$ , the correct answer is that there is no such $x$ for which $f^{-1}(x) = 2$ .
The correct answer is:
$\boxed{\text{E. None}}$

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QuestionA. -2 B. 2 C. 0 D. $\infty$ E.None Q.33) Let $f(x)=2 x^{2}+3$ find the value of $x$ such that $f^{-1}(x)=2$ A. 19 B. 8 C. -19 D. 2 E.None

Studdy Solution

STEP 1

1. We are given the function $f(x) = 2x^2 + 3$ .
2. We need to find the value of $x$ such that $f^{-1}(x) = 2$ .

STEP 2

1. Understand the meaning of $f^{-1}(x) = 2$ .
2. Set up the equation based on the inverse function.
3. Solve the equation for $x$ .
4. Verify the solution.

STEP 3

Understand that $f^{-1}(x) = 2$ means that when $f(x) = y$ , then $f^{-1}(y) = x$ . Therefore, we need to find $y$ such that $f(y) = 2$ .

STEP 4

Set up the equation $f(y) = 2$ :
$2y^2 + 3 = 2$

STEP 5

Solve the equation for $y$ :
$2y^2 + 3 = 2$ $2y^2 = 2 - 3$ $2y^2 = -1$

STEP 6

Since $2y^2 = -1$ results in a negative number on the right side, there is no real number solution for $y$ .

STEP 7

Verify the solution. Since there is no real solution for $y$ , the correct answer is that there is no such $x$ for which $f^{-1}(x) = 2$ .
The correct answer is:
$\boxed{\text{E. None}}$

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QuestionA. -2 B. 2 C. 0 D. ∞\infty∞ E.None Q.33) Let f(x)=2x2+3f(x)=2 x^{2}+3f(x)=2x2+3 find the value of xxx such that f−1(x)=2f^{-1}(x)=2f−1(x)=2 A. 19 B. 8 C. -19 D. 2 E.None

Studdy Solution

STEP 1

1. We are given the function f(x)=2x2+3 f(x) = 2x^2 + 3 f(x)=2x2+3.2. We need to find the value of x x x such that f−1(x)=2 f^{-1}(x) = 2 f−1(x)=2.

STEP 2

1. Understand the meaning of f−1(x)=2 f^{-1}(x) = 2 f−1(x)=2.2. Set up the equation based on the inverse function.3. Solve the equation for x x x.4. Verify the solution.

STEP 3

Understand that f−1(x)=2 f^{-1}(x) = 2 f−1(x)=2 means that when f(x)=y f(x) = y f(x)=y, then f−1(y)=x f^{-1}(y) = x f−1(y)=x. Therefore, we need to find y y y such that f(y)=2 f(y) = 2 f(y)=2.

STEP 4

Set up the equation f(y)=2 f(y) = 2 f(y)=2:2y2+3=2 2y^2 + 3 = 2 2y2+3=2

STEP 5

Solve the equation for y y y:2y2+3=2 2y^2 + 3 = 2 2y2+3=2 2y2=2−3 2y^2 = 2 - 3 2y2=2−3 2y2=−1 2y^2 = -1 2y2=−1

STEP 6

Since 2y2=−1 2y^2 = -1 2y2=−1 results in a negative number on the right side, there is no real number solution for y y y.

STEP 7

Verify the solution. Since there is no real solution for y y y, the correct answer is that there is no such x x x for which f−1(x)=2 f^{-1}(x) = 2 f−1(x)=2. The correct answer is:E. None \boxed{\text{E. None}} E. None​

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QuestionA. -2 B. 2 C. 0 D. $\infty$ E.None Q.33) Let $f(x)=2 x^{2}+3$ find the value of $x$ such that $f^{-1}(x)=2$ A. 19 B. 8 C. -19 D. 2 E.None

1. We are given the function $f(x) = 2x^2 + 3$ .
2. We need to find the value of $x$ such that $f^{-1}(x) = 2$ .

1. Understand the meaning of $f^{-1}(x) = 2$ .
2. Set up the equation based on the inverse function.
3. Solve the equation for $x$ .
4. Verify the solution.

Understand that $f^{-1}(x) = 2$ means that when $f(x) = y$ , then $f^{-1}(y) = x$ . Therefore, we need to find $y$ such that $f(y) = 2$ .

Set up the equation $f(y) = 2$ :
$2y^2 + 3 = 2$

Solve the equation for $y$ :
$2y^2 + 3 = 2$ $2y^2 = 2 - 3$ $2y^2 = -1$

Since $2y^2 = -1$ results in a negative number on the right side, there is no real number solution for $y$ .

Verify the solution. Since there is no real solution for $y$ , the correct answer is that there is no such $x$ for which $f^{-1}(x) = 2$ .
The correct answer is:
$\boxed{\text{E. None}}$