Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

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Step 1: Understand the function and its domain. The function given is $ f(x) = 3x^2 + 1 $ with the domain $ x \leq 0 $. This means we are only considering the left half of the parabola.

Step 2: Replace $ f(x) $ with $ y $ to make it easier to work with: $ y = 3x^2 + 1 $.

Step 3: Solve for $ x $ in terms of $ y $. Start by isolating the $ x^2 $ term: $ y - 1 = 3x^2 $.

Step 4: Divide both sides by 3 to further isolate $ x^2 $: $ \frac{y - 1}{3} = x^2 $.

Step 5: Since $ x \leq 0 $, take the negative square root to solve for $ x $: $ x = -\sqrt{\frac{y - 1}{3}} $. This expression represents the inverse function, $ f^{-1}(y) $.

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Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). When finding an inverse function, it is crucial to consider the domain of the original function, as it affects the range of the inverse function. In this case, the restriction x ≤ 0 is important for determining the inverse.

Solving for x

To find the inverse of a function, one typically starts by replacing f(x) with y, then solving for x in terms of y. This often involves algebraic manipulation, such as isolating x on one side of the equation. Once x is expressed in terms of y, the inverse function can be written as f⁻¹(y) = x, and then it can be rewritten in terms of x.

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교과서 질문

The National Weather Service releases approximately $70,000$ radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about $1000$ ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point $6$ ft above the ground and that $5$ seconds later, it is $83$ ft above the ground. Let $f$\left$(t$\right$)$ represent the height (in feet) that the radiosonde is above the ground $t$ seconds after it is released. Evaluate $\frac{f\left(5\right)-f\left(0\right)}{5-0}$ and interpret the meaning of this quotient.

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교과서 질문

Solve each equation.

$7^{y-3}=50$

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교과서 질문

Solving equations Solve the following equations.

7ˣ = 21

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교과서 질문

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x^{4} + 5 x^{2} - 12$