1. **Problem statement:** We are given four functions and need to analyze or simplify them as needed. 2. **Function a:** $f(x) = x^6 + 2x - 1$ - This is a polynomial function with terms $x^6$, $2x$, and a constant $-1$. - No simplification needed. 3. **Function b:** $h(x) = \frac{3}{5} x^5 - 4x^{-2}$ - Rewrite negative exponent: $x^{-2} = \frac{1}{x^2}$. - So, $h(x) = \frac{3}{5} x^5 - \frac{4}{x^2}$. 4. **Function c:** $f(t) = \sqrt{t} + \frac{1}{t^3}$ - Rewrite square root as exponent: $\sqrt{t} = t^{\frac{1}{2}}$. - So, $f(t) = t^{\frac{1}{2}} + t^{-3}$. 5. **Function d:** $f(x) = x^{7+1} + \frac{1}{x^7}$ - Simplify exponent: $7+1=8$. - So, $f(x) = x^8 + x^{-7}$. **Summary:** - a) $f(x) = x^6 + 2x - 1$ - b) $h(x) = \frac{3}{5} x^5 - \frac{4}{x^2}$ - c) $f(t) = t^{\frac{1}{2}} + t^{-3}$ - d) $f(x) = x^8 + x^{-7}$ These are the simplified or rewritten forms of the given functions.