19. is the inverse of the function $f(x) = -60(x - 3)^2 - 50$ also a fu…

Question

is the inverse of the function $f(x) = -60(x - 3)^2 - 50$ also a function? explain your thought process. (2 marks)

Solution Steps

Understand the question
1. is the inverse of the function $f(x) = -60(x - 3)^2 - 50$ also a function? explain your thought process. (2 marks)
Explanation
Step1: Identify function type

The given function $f(x) = -60(x-3)^2 - 50$ is a quadratic function, which graphs as a downward-opening parabola (since the coefficient of the squared term is negative).

Step2: Apply Horizontal Line Test

For a relation to be a function, every input has exactly one output. To check if an inverse is a function, we use the horizontal line test on the original function: if any horizontal line intersects the graph more than once, the inverse is not a function.
A parabola (opening up or down) will be intersected by most horizontal lines twice, meaning multiple $x$-values map to the same $y$-value.

Step3: Verify algebraically

Solve for the inverse relation:
1. Let $y = -60(x-3)^2 - 50$
2. Swap $x$ and $y$: $x = -60(y-3)^2 - 50$
3. Isolate the squared term:
$x + 50 = -60(y-3)^2$
$\frac{x + 50}{-60} = (y-3)^2$
1. Take square root: $y - 3 = \pm\sqrt{\frac{-(x + 50)}{60}}$
$y = 3 \pm\sqrt{\frac{-(x + 50)}{60}}$
The $\pm$ shows that for valid $x$-values, there are two corresponding $y$-values, so the inverse is not a function.
Final answer
No, the inverse of $f(x) = -60(x-3)^2 - 50$ is not a function. The original function is a downward-opening parabola, which fails the horizontal line test (horizontal lines intersect the parabola twice). Algebraically, solving for the inverse results in a $\pm$ sign, meaning a single input maps to two outputs, violating the definition of a function.

Answer

Explanation

Step1: Identify function type

The given function $f(x) = -60(x-3)^2 - 50$ is a quadratic function, which graphs as a downward-opening parabola (since the coefficient of the squared term is negative).

Step2: Apply Horizontal Line Test

For a relation to be a function, every input has exactly one output. To check if an inverse is a function, we use the horizontal line test on the original function: if any horizontal line intersects the graph more than once, the inverse is not a function.
A parabola (opening up or down) will be intersected by most horizontal lines twice, meaning multiple $x$-values map to the same $y$-value.

Step3: Verify algebraically

Solve for the inverse relation:

Let $y = -60(x-3)^2 - 50$
Swap $x$ and $y$: $x = -60(y-3)^2 - 50$
Isolate the squared term:

$x + 50 = -60(y-3)^2$
$\frac{x + 50}{-60} = (y-3)^2$

Take square root: $y - 3 = \pm\sqrt{\frac{-(x + 50)}{60}}$

$y = 3 \pm\sqrt{\frac{-(x + 50)}{60}}$
The $\pm$ shows that for valid $x$-values, there are two corresponding $y$-values, so the inverse is not a function.

Answer

No, the inverse of $f(x) = -60(x-3)^2 - 50$ is not a function. The original function is a downward-opening parabola, which fails the horizontal line test (horizontal lines intersect the parabola twice). Algebraically, solving for the inverse results in a $\pm$ sign, meaning a single input maps to two outputs, violating the definition of a function.

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Question Analysis

Subject mathematics

Sub Subject calculus

Education Level high school

Difficulty unspecified

Question Type text only

Multi Question No

Question Count 1

Analysis Status completed

Analyzed At 2026-02-09T20:08:12

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19. is the inverse of the function $f(x) = -60(x - 3)^2 - 50$ also a function? explain your thought process. (2 marks)

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