Solve f(x)^-1=x^2+7

$$f ( x ) ^ { - 1 } = x ^ { 2 } + 7$$

Solve for f

$$f=x\left(x^{2}+7\right)$$

$$x\neq 0$$

Solution Steps

Reorder the terms.

$$\frac{1}{x}f=x^{2}+7$$

Multiply both sides of the equation by x.

$$1f=xx^{2}+x\times 7$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$1f=x^{3}+x\times 7$$

Reorder the terms.

$$f=x^{3}+7x$$

Solve for x

$$x=\frac{2^{\frac{2}{3}}\sqrt[3]{\frac{\sqrt{81f^{2}+4116}}{9}+f}+2^{\frac{2}{3}}\sqrt[3]{-\frac{\sqrt{81f^{2}+4116}}{9}+f}}{2}$$

$$f\neq 0$$