Question

Solve for the inverse of the dilated and translated quadratic function f(x)=7x2+5
𝑓
(
𝑥
)
=
7
𝑥
2
+
5
with domain restricted to x≥0
𝑥
≥
0
.(1 point)
Responses

f−1(x)=x−57−−−√
𝑓
−
1
(
𝑥
)
=
𝑥
−
5
7
f inverse left parenthesis x right parenthesis equals Start Root Start Fraction x minus 5 over 7 End Fraction End Root

f−1(x)=x+57−−−√
𝑓
−
1
(
𝑥
)
=
𝑥
+
5
7
f inverse left parenthesis x right parenthesis equals Start Root Start Fraction x plus 5 over 7 End Fraction End Root

f−1(x)=7x−−√+5
𝑓
−
1
(
𝑥
)
=
7
𝑥
+
5
f inverse left parenthesis x right parenthesis equals Start Root 7 x End Root plus 5

f−1(x)=7x−5−−−−−√

Answer 1

To find the inverse of the function \( f(x) = 7x^2 + 5 \) where \( x \geq 0 \), we will follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = 7x^2 + 5
\]

2. **Solve for \( x \)** in terms of \( y \):
- First, subtract 5 from both sides:
\[
y - 5 = 7x^2
\]

- Next, divide by 7:
\[
\frac{y - 5}{7} = x^2
\]

- Now take the square root of both sides:
\[
x = \sqrt{\frac{y - 5}{7}}
\]
Since \( x \geq 0 \), we take the positive square root.

3. **Express the inverse function**:
- Swap \( x \) and \( y \):
\[
f^{-1}(x) = \sqrt{\frac{x - 5}{7}}
\]

Thus, the inverse function is:

\[
f^{-1}(x) = \sqrt{\frac{x - 5}{7}}
\]

In terms of the possible responses given, it matches with:

\[
f^{-1}(x) = \sqrt{\frac{x - 5}{7}}
\]

Therefore, the correct answer is:

\[
f^{-1}(x) = \sqrt{\frac{x - 5}{7}}
\]